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# -*- coding: utf-8 -*- r""" The Steenrod algebra
AUTHORS:
- John H. Palmieri (2008-07-30): version 0.9: Initial implementation. - John H. Palmieri (2010-06-30): version 1.0: Implemented sub-Hopf algebras and profile functions; direct multiplication of admissible sequences (rather than conversion to the Milnor basis); implemented the Steenrod algebra using CombinatorialFreeModule; improved the test suite.
This module defines the mod `p` Steenrod algebra `\mathcal{A}_p`, some of its properties, and ways to define elements of it.
From a topological point of view, `\mathcal{A}_p` is the algebra of stable cohomology operations on mod `p` cohomology; thus for any topological space `X`, its mod `p` cohomology algebra `H^*(X,\mathbf{F}_p)` is a module over `\mathcal{A}_p`.
From an algebraic point of view, `\mathcal{A}_p` is an `\mathbf{F}_p`-algebra; when `p=2`, it is generated by elements `\text{Sq}^i` for `i\geq 0` (the *Steenrod squares*), and when `p` is odd, it is generated by elements `\mathcal{P}^i` for `i \geq 0` (the *Steenrod reduced pth powers*) along with an element `\beta` (the *mod p Bockstein*). The Steenrod algebra is graded: `\text{Sq}^i` is in degree `i` for each `i`, `\beta` is in degree 1, and `\mathcal{P}^i` is in degree `2(p-1)i`.
The unit element is `\text{Sq}^0` when `p=2` and `\mathcal{P}^0` when `p` is odd. The generating elements also satisfy the *Adem relations*. At the prime 2, these have the form
.. MATH::
\text{Sq}^a \text{Sq}^b = \sum_{c=0}^{[a/2]} \binom{b-c-1}{a-2c} \text{Sq}^{a+b-c} \text{Sq}^c.
At odd primes, they are a bit more complicated; see Steenrod and Epstein [SE1962]_ or :mod:`sage.algebras.steenrod.steenrod_algebra_bases` for full details. These relations lead to the existence of the *Serre-Cartan* basis for `\mathcal{A}_p`.
The mod `p` Steenrod algebra has the structure of a Hopf algebra, and Milnor [Mil1958]_ has a beautiful description of the dual, leading to a construction of the *Milnor basis* for `\mathcal{A}_p`. In this module, elements in the Steenrod algebra are represented, by default, using the Milnor basis.
.. rubric:: Bases for the Steenrod algebra
There are a handful of other bases studied in the literature; the paper by Monks [Mon1998]_ is a good reference. Here is a quick summary:
- The *Milnor basis*. When `p=2`, the Milnor basis consists of symbols of the form `\text{Sq}(m_1, m_2, ..., m_t)`, where each `m_i` is a non-negative integer and if `t>1`, then the last entry `m_t > 0`. When `p` is odd, the Milnor basis consists of symbols of the form `Q_{e_1} Q_{e_2} ... \mathcal{P}(m_1, m_2, ..., m_t)`, where `0 \leq e_1 < e_2 < ...`, each `m_i` is a non-negative integer, and if `t>1`, then the last entry `m_t > 0`.
When `p=2`, it can be convenient to use the notation `\mathcal{P}(-)` to mean `\text{Sq}(-)`, so that there is consistent notation for all primes.
- The *Serre-Cartan basis*. This basis consists of 'admissible monomials' in the Steenrod operations. Thus at the prime 2, it consists of monomials `\text{Sq}^{m_1} \text{Sq}^{m_2} ... \text{Sq}^{m_t}` with `m_i \geq 2m_{i+1}` for each `i`. At odd primes, this basis consists of monomials `\beta^{\epsilon_0} \mathcal{P}^{s_1} \beta^{\epsilon_1} \mathcal{P}^{s_2} ... \mathcal{P}^{s_k} \beta^{\epsilon_k}` with each `\epsilon_i` either 0 or 1, `s_i \geq p s_{i+1} + \epsilon_i`, and `s_k \geq 1`.
Most of the rest of the bases are only defined when `p=2`. The only exceptions are the `P^s_t`-bases and the commutator bases, which are defined at all primes.
- *Wood's Y basis*. For pairs of non-negative integers `(m,k)`, let `w(m,k) = \text{Sq}^{2^m (2^{k+1}-1)}`. Wood's `Y` basis consists of monomials `w(m_0,k_0) ... w(m_t, k_t)` with `(m_i,k_i) > (m_{i+1},k_{i+1})`, in left lex order.
- *Wood's Z basis*. For pairs of non-negative integers `(m,k)`, let `w(m,k) = \text{Sq}^{2^m (2^{k+1}-1)}`. Wood's `Z` basis consists of monomials `w(m_0,k_0) ... w(m_t, k_t)` with `(m_i+k_i,m_i) > (m_{i+1}+k_{i+1},m_{i+1})`, in left lex order.
- *Wall's basis*. For any pair of integers `(m,k)` with `m \geq k \geq 0`, let `Q^m_k = \text{Sq}^{2^k} \text{Sq}^{2^{k+1}} ... \text{Sq}^{2^m}`. The elements of Wall's basis are monomials `Q^{m_0}_{k_0} ... Q^{m_t}_{k_t}` with `(m_i, k_i) > (m_{i+1}, k_{i+1})`, ordered left lexicographically.
(Note that `Q^m_k` is the reverse of the element `X^m_k` used in defining Arnon's A basis.)
- *Arnon's A basis*. For any pair of integers `(m,k)` with `m \geq k \geq 0`, let `X^m_k = \text{Sq}^{2^m} \text{Sq}^{2^{m-1}} ... \text{Sq}^{2^k}`. The elements of Arnon's A basis are monomials `X^{m_0}_{k_0} ... X^{m_t}_{k_t}` with `(m_i, k_i) < (m_{i+1}, k_{i+1})`, ordered left lexicographically.
(Note that `X^m_k` is the reverse of the element `Q^m_k` used in defining Wall's basis.)
- *Arnon's C basis*. The elements of Arnon's C basis are monomials of the form `\text{Sq}^{t_1} ... \text{Sq}^{t_m}` where for each `i`, we have `t_i \leq 2t_{i+1}` and `2^i | t_{m-i}`.
- `P^s_t` *bases*. Let `p=2`. For integers `s \geq 0` and `t > 0`, the element `P^s_t` is the Milnor basis element `\mathcal{P}(0, ..., 0, p^s, 0, ...)`, with the nonzero entry in position `t`. To obtain a `P^s_t`-basis, for each set `\{P^{s_1}_{t_1}, ..., P^{s_k}_{t_k}\}` of (distinct) `P^s_t`'s, one chooses an ordering and forms the monomials
.. MATH::
(P^{s_1}_{t_1})^{i_1} ... (P^{s_k}_{t_k})^{i_k}
for all exponents `i_j` with `0 < i_j < p`. When `p=2`, the set of all such monomials then forms a basis, and when `p` is odd, if one multiplies each such monomial on the left by products of the form `Q_{e_1} Q_{e_2} ...` with `0 \leq e_1 < e_2 < ...`, one obtains a basis.
Thus one gets a basis by choosing an ordering on each set of `P^s_t`'s. There are infinitely many orderings possible, and we have implemented four of them:
- 'rlex': right lexicographic ordering
- 'llex': left lexicographic ordering
- 'deg': ordered by degree, which is the same as left lexicographic ordering on the pair `(s+t,t)`
- 'revz': left lexicographic ordering on the pair `(s+t,s)`, which is the reverse of the ordering used (on elements in the same degrees as the `P^s_t`'s) in Wood's Z basis: 'revz' stands for 'reversed Z'. This is the default: 'pst' is the same as 'pst_revz'.
- *Commutator bases*. Let `c_{i,1} = \mathcal{P}(p^i)`, let `c_{i,2} = [c_{i+1,1}, c_{i,1}]`, and inductively define `c_{i,k} = [c_{i+k-1,1}, c_{i,k-1}]`. Thus `c_{i,k}` is a `k`-fold iterated commutator of the elements `\mathcal{P}(p^i)`, ..., `\mathcal{P}(p^{i+k-1})`. Note that `\dim c_{i,k} = \dim P^i_k`.
Commutator bases are obtained in much the same way as `P^s_t`-bases: for each set `\{c_{s_1,t_1}, ..., c_{s_k,t_k}\}` of (distinct) `c_{s,t}`'s, one chooses an ordering and forms the resulting monomials
.. MATH::
c_{s_1, t_1}^{i_1} ... c_{s_k,t_k}^{i_k}
for all exponents `i_j` with `0 < i_j < p`. When `p` is odd, one also needs to left-multiply by products of the `Q_i`'s. As for `P^s_t`-bases, every ordering on each set of iterated commutators determines a basis, and the same four orderings have been defined for these bases as for the `P^s_t` bases: 'rlex', 'llex', 'deg', 'revz'.
.. rubric:: Sub-Hopf algebras of the Steenrod algebra
The sub-Hopf algebras of the Steenrod algebra have been classified. Milnor proved that at the prime 2, the dual of the Steenrod algebra `A_*` is isomorphic to a polynomial algebra
.. MATH::
A_* \cong \GF{2} [\xi_1, \xi_2, \xi_3, ...].
The Milnor basis is dual to the monomial basis. Furthermore, any sub-Hopf algebra corresponds to a quotient of this of the form
.. MATH::
A_* /(\xi_1^{2^{e_1}}, \xi_2^{2^{e_2}}, \xi_3^{2^{e_3}}, ...).
The list of exponents `(e_1, e_2, ...)` may be considered a function `e` from the positive integers to the extended non-negative integers (the non-negative integers and `\infty`); this is called the *profile function* for the sub-Hopf algebra. The profile function must satisfy the condition
- `e(r) \geq \min( e(r-i) - i, e(i))` for all `0 < i < r`.
At odd primes, the situation is similar: the dual is isomorphic to the tensor product of a polynomial algebra and an exterior algebra,
.. MATH::
A_* = \GF{p} [\xi_1, \xi_2, \xi_3, ...] \otimes \Lambda (\tau_0, \tau_1, ...),
and any sub-Hopf algebra corresponds to a quotient of this of the form
.. MATH::
A_* / (\xi_1^{p^{e_1}}, \xi_2^{p^{e_2}}, ...; \tau_0^{k_0}, \tau_1^{k_1}, ...).
Here the profile function has two pieces, `e` as at the prime 2, and `k`, which maps the non-negative integers to the set `\{1, 2\}`. These must satisfy the following conditions:
- `e(r) \geq \min( e(r-i) - i, e(i))` for all `0 < i < r`.
- if `k(i+j) = 1`, then either `e(i) \leq j` or `k(j) = 1` for all `i \geq 1`, `j \geq 0`.
(See Adams-Margolis [AM1974]_, for example, for these results on profile functions.)
This module allows one to construct the Steenrod algebra or any of its sub-Hopf algebras, at any prime. When defining a sub-Hopf algebra, you must work with the Milnor basis or a `P^s_t`-basis.
.. rubric:: Elements of the Steenrod algebra
Basic arithmetic, `p=2`. To construct an element of the mod 2 Steenrod algebra, use the function ``Sq``::
sage: a = Sq(1,2) sage: b = Sq(4,1) sage: z = a + b sage: z Sq(1,2) + Sq(4,1) sage: Sq(4) * Sq(1,2) Sq(1,1,1) + Sq(2,3) + Sq(5,2) sage: z**2 # non-negative exponents work as they should Sq(1,2,1) + Sq(4,1,1) sage: z**0 1
Basic arithmetic, `p>2`. To construct an element of the mod `p` Steenrod algebra when `p` is odd, you should first define a Steenrod algebra, using the ``SteenrodAlgebra`` command::
sage: A3 = SteenrodAlgebra(3)
Having done this, the newly created algebra ``A3`` has methods ``Q`` and ``P`` which construct elements of ``A3``::
sage: c = A3.Q(1,3,6); c Q_1 Q_3 Q_6 sage: d = A3.P(2,0,1); d P(2,0,1) sage: c * d Q_1 Q_3 Q_6 P(2,0,1) sage: e = A3.P(3) sage: d * e P(5,0,1) sage: e * d P(1,1,1) + P(5,0,1) sage: c * c 0 sage: e ** 3 2 P(1,2)
Note that one can construct an element like ``c`` above in one step, without first constructing the algebra::
sage: c = SteenrodAlgebra(3).Q(1,3,6) sage: c Q_1 Q_3 Q_6
And of course, you can do similar constructions with the mod 2 Steenrod algebra::
sage: A = SteenrodAlgebra(2); A mod 2 Steenrod algebra, milnor basis sage: A.Sq(2,3,5) Sq(2,3,5) sage: A.P(2,3,5) # when p=2, P = Sq Sq(2,3,5) sage: A.Q(1,4) # when p=2, this gives a product of Milnor primitives Sq(0,1,0,0,1)
Associated to each element is its prime (the characteristic of the underlying base field) and its basis (the basis for the Steenrod algebra in which it lies)::
sage: a = SteenrodAlgebra(basis='milnor').Sq(1,2,1) sage: a.prime() 2 sage: a.basis_name() 'milnor' sage: a.degree() 14
It can be viewed in other bases::
sage: a.milnor() # same as a Sq(1,2,1) sage: a.change_basis('adem') Sq^9 Sq^4 Sq^1 + Sq^11 Sq^2 Sq^1 + Sq^13 Sq^1 sage: a.change_basis('adem').change_basis('milnor') Sq(1,2,1)
Regardless of the prime, each element has an ``excess``, and if the element is homogeneous, a ``degree``. The excess of `\text{Sq}(i_1,i_2,i_3,...)` is `i_1 + i_2 + i_3 + ...`; when `p` is odd, the excess of `Q_{0}^{e_0} Q_{1}^{e_1} ... \mathcal{P}(r_1, r_2, ...)` is `\sum e_i + 2 \sum r_i`. The excess of a linear combination of Milnor basis elements is the minimum of the excesses of those basis elements.
The degree of `\text{Sq}(i_1,i_2,i_3,...)` is `\sum (2^n-1) i_n`, and when `p` is odd, the degree of `Q_{0}^{\epsilon_0} Q_{1}^{\epsilon_1} ... \mathcal{P}(r_1, r_2, ...)` is `\sum \epsilon_i (2p^i - 1) + \sum r_j (2p^j - 2)`. The degree of a linear combination of such terms is only defined if the terms all have the same degree.
Here are some simple examples::
sage: z = Sq(1,2) + Sq(4,1) sage: z.degree() 7 sage: (Sq(0,0,1) + Sq(5,3)).degree() Traceback (most recent call last): ... ValueError: Element is not homogeneous. sage: Sq(7,2,1).excess() 10 sage: z.excess() 3 sage: B = SteenrodAlgebra(3) sage: x = B.Q(1,4) sage: y = B.P(1,2,3) sage: x.degree() 166 sage: x.excess() 2 sage: y.excess() 12
Elements have a ``weight`` in the May filtration, which (when `p=2`) is related to the ``height`` function defined by Wall::
sage: Sq(2,1,5).may_weight() 9 sage: Sq(2,1,5).wall_height() [2, 3, 2, 1, 1] sage: b = Sq(4)*Sq(8) + Sq(8)*Sq(4) sage: b.may_weight() 2 sage: b.wall_height() [0, 0, 1, 1]
Odd primary May weights::
sage: A5 = SteenrodAlgebra(5) sage: a = A5.Q(1,2,4) sage: b = A5.P(1,2,1) sage: a.may_weight() 10 sage: b.may_weight() 8 sage: (a * b).may_weight() 18 sage: A5.P(0,0,1).may_weight() 3
Since the Steenrod algebra is a Hopf algebra, every element has a coproduct and an antipode::
sage: Sq(5).coproduct() 1 # Sq(5) + Sq(1) # Sq(4) + Sq(2) # Sq(3) + Sq(3) # Sq(2) + Sq(4) # Sq(1) + Sq(5) # 1 sage: Sq(5).antipode() Sq(2,1) + Sq(5) sage: d = Sq(0,0,1); d Sq(0,0,1) sage: d.antipode() Sq(0,0,1) sage: Sq(4).antipode() Sq(1,1) + Sq(4) sage: (Sq(4) * Sq(2)).antipode() Sq(6) sage: SteenrodAlgebra(7).P(3,1).antipode() P(3,1)
Applying the antipode twice returns the original element::
sage: y = Sq(8)*Sq(4) sage: y == (y.antipode()).antipode() True
Internal representation: you can use any element as an iterator (``for x in a: ...``), and the method :meth:`monomial_coefficients` returns a dictionary with keys tuples representing basis elements and with corresponding value representing the coefficient of that term::
sage: c = Sq(5).antipode(); c Sq(2,1) + Sq(5) sage: for mono, coeff in c: print((coeff, mono)) (1, (5,)) (1, (2, 1)) sage: c.monomial_coefficients() {(2, 1): 1, (5,): 1} sage: sorted(c.monomials(), key=lambda x: x.support()) [Sq(2,1), Sq(5)] sage: sorted(c.support()) [(2, 1), (5,)] sage: Adem = SteenrodAlgebra(basis='adem') sage: elt = Adem.Sq(10) + Adem.Sq(9) * Adem.Sq(1) sage: sorted(elt.monomials(), key=lambda x: x.support()) [Sq^9 Sq^1, Sq^10]
sage: A7 = SteenrodAlgebra(p=7) sage: a = A7.P(1) * A7.P(1); a 2 P(2) sage: a.leading_coefficient() 2 sage: a.leading_monomial() P(2) sage: a.leading_term() 2 P(2) sage: a.change_basis('adem').monomial_coefficients() {(0, 2, 0): 2}
The tuple in the previous output stands for the element `\beta^0 P^2 \beta^0`, i.e., `P^2`. Going in the other direction, if you want to specify a basis element by giving the corresponding tuple, you can use the :meth:`monomial` method on the algebra::
sage: SteenrodAlgebra(p=7, basis='adem').monomial((0, 2, 0)) P^2 sage: 10 * SteenrodAlgebra(p=7, basis='adem').monomial((0, 2, 0)) 3 P^2
In the following example, elements in Wood's Z basis are certain products of the elements `w(m,k) = \text{Sq}^{2^m (2^{k+1}-1)}`. Internally, each `w(m,k)` is represented by the pair `(m,k)`, and products of them are represented by tuples of such pairs. ::
sage: A = SteenrodAlgebra(basis='wood_z') sage: t = ((2, 0), (0, 0)) sage: A.monomial(t) Sq^4 Sq^1
See the documentation for :func:`SteenrodAlgebra` for more details and examples. """
#***************************************************************************** # Copyright (C) 2008-2010 John H. Palmieri <palmieri@math.washington.edu> # Distributed under the terms of the GNU General Public License (GPL) #***************************************************************************** from __future__ import print_function, absolute_import from six.moves import range
from sage.combinat.free_module import CombinatorialFreeModule from sage.misc.lazy_attribute import lazy_attribute from sage.misc.cachefunc import cached_method from sage.categories.all import ModulesWithBasis, tensor, Hom
###################################################### # the main class ######################################################
class SteenrodAlgebra_generic(CombinatorialFreeModule): r""" The mod `p` Steenrod algebra.
Users should not call this, but use the function :func:`SteenrodAlgebra` instead. See that function for extensive documentation.
EXAMPLES::
sage: sage.algebras.steenrod.steenrod_algebra.SteenrodAlgebra_generic() mod 2 Steenrod algebra, milnor basis sage: sage.algebras.steenrod.steenrod_algebra.SteenrodAlgebra_generic(5) mod 5 Steenrod algebra, milnor basis sage: sage.algebras.steenrod.steenrod_algebra.SteenrodAlgebra_generic(5, 'adem') mod 5 Steenrod algebra, serre-cartan basis """ @staticmethod def __classcall__(self, p=2, basis='milnor', **kwds): """ This normalizes the basis name and the profile, to make unique representation work properly.
EXAMPLES::
sage: SteenrodAlgebra(basis='adem') is SteenrodAlgebra(basis='serre-cartan') True sage: SteenrodAlgebra(profile=[3,2,1,0]) is SteenrodAlgebra(profile=lambda n: max(4-n,0), truncation_type=0) True sage: SteenrodAlgebra(p=5) is SteenrodAlgebra(p=5, generic=True) True """ else: raise ValueError("option 'generic' is not a boolean")
truncation_type=std_type, generic=std_generic)
def __init__(self, p=2, basis='milnor', **kwds): r""" INPUT:
- ``p`` - positive prime integer (optional, default 2) - ``basis`` - string (optional, default = 'milnor') - ``profile`` - profile function (optional, default ``None``) - ``truncation_type`` - (optional, default 'auto') - ``precision`` - (optional, default ``None``) - ``generic`` - (optional, default 'auto')
OUTPUT: mod `p` Steenrod algebra with basis, or a sub-Hopf algebra of the mod `p` Steenrod algebra defined by the given profile function.
See :func:`SteenrodAlgebra` for full documentation.
EXAMPLES::
sage: SteenrodAlgebra() # 2 is the default prime mod 2 Steenrod algebra, milnor basis sage: SteenrodAlgebra(5) mod 5 Steenrod algebra, milnor basis sage: SteenrodAlgebra(2, 'milnor').Sq(0,1) Sq(0,1) sage: SteenrodAlgebra(2, 'adem').Sq(0,1) Sq^2 Sq^1 + Sq^3
TESTS::
sage: TestSuite(SteenrodAlgebra()).run() sage: TestSuite(SteenrodAlgebra(profile=[4,3,2,2,1])).run() sage: TestSuite(SteenrodAlgebra(basis='adem')).run() sage: TestSuite(SteenrodAlgebra(basis='wall')).run() sage: TestSuite(SteenrodAlgebra(basis='arnonc')).run() # long time sage: TestSuite(SteenrodAlgebra(basis='woody')).run() # long time sage: A3 = SteenrodAlgebra(3) sage: A3.category() Category of graded hopf algebras with basis over Finite Field of size 3 sage: TestSuite(A3).run() sage: TestSuite(SteenrodAlgebra(basis='adem', p=3)).run() sage: TestSuite(SteenrodAlgebra(basis='pst_llex', p=7)).run() # long time sage: TestSuite(SteenrodAlgebra(basis='comm_deg', p=5)).run() # long time sage: TestSuite(SteenrodAlgebra(p=2,generic=True)).run() """
raise ValueError("%s is not prime." % p) or (self._generic and profile != ((), ()) and len(profile[0]) > 0 and profile[0][0] < Infinity) or (truncation_type < Infinity)): category=basis_category, name = "basis key family of %s" % self, cache = False)
p=p, basis=basis, profile=profile, truncation_type=truncation_type, generic=self._generic)
base_ring, basis_set, prefix=self._basis_name, element_class=self.Element, category = GradedHopfAlgebrasWithBasis(base_ring), scalar_mult = ' ')
def _basis_key_iterator(self): """ An iterator for the basis keys of the Steenrod algebra.
EXAMPLES::
sage: A = SteenrodAlgebra(3,basis='adem') sage: for (idx,key) in zip((1,..,10),A._basis_key_iterator()): ....: print("> %2d %-20s %s" % (idx,key,A.monomial(key))) > 1 () 1 > 2 (1,) beta > 3 (0, 1, 0) P^1 > 4 (1, 1, 0) beta P^1 > 5 (0, 1, 1) P^1 beta > 6 (1, 1, 1) beta P^1 beta > 7 (0, 2, 0) P^2 > 8 (1, 2, 0) beta P^2 > 9 (0, 2, 1) P^2 beta > 10 (1, 2, 1) beta P^2 beta """ else: p=self.prime(), basis=self._basis_name, profile=self._profile, truncation_type=self._truncation_type)
def prime(self): r""" The prime associated to self.
EXAMPLES::
sage: SteenrodAlgebra(p=2, profile=[1,1]).prime() 2 sage: SteenrodAlgebra(p=7).prime() 7 """
def basis_name(self): r""" The basis name associated to self.
EXAMPLES::
sage: SteenrodAlgebra(p=2, profile=[1,1]).basis_name() 'milnor' sage: SteenrodAlgebra(basis='serre-cartan').basis_name() 'serre-cartan' sage: SteenrodAlgebra(basis='adem').basis_name() 'serre-cartan' """
def _has_nontrivial_profile(self): r""" True if the profile function for this algebra seems to be that for a proper sub-Hopf algebra of the Steenrod algebra.
EXAMPLES::
sage: SteenrodAlgebra()._has_nontrivial_profile() False sage: SteenrodAlgebra(p=3)._has_nontrivial_profile() False sage: SteenrodAlgebra(profile=[3,2,1])._has_nontrivial_profile() True sage: SteenrodAlgebra(profile=([1], [2, 2]), p=3)._has_nontrivial_profile() True sage: SteenrodAlgebra(generic=True)._has_nontrivial_profile() False sage: SteenrodAlgebra(generic=True, profile=[[3,2,1], []])._has_nontrivial_profile() True
Check that a bug in :trac:`11832` has been fixed::
sage: P3 = SteenrodAlgebra(p=3, profile=(lambda n: Infinity, lambda n: 1)) sage: P3._has_nontrivial_profile() True """ and profile[0] < Infinity) or (trunc < Infinity)) ((len(profile[0]) > 0 and profile[0][0] < Infinity) or (len(profile[1]) > 0 and min(profile[1]) == 1))) or (trunc < Infinity))
def _repr_(self): r""" Printed representation of the Steenrod algebra.
EXAMPLES::
sage: SteenrodAlgebra(3) mod 3 Steenrod algebra, milnor basis sage: SteenrodAlgebra(2, basis='adem') mod 2 Steenrod algebra, serre-cartan basis sage: B = SteenrodAlgebra(2003) sage: B._repr_() 'mod 2003 Steenrod algebra, milnor basis' sage: SteenrodAlgebra(generic=True, basis='adem') generic mod 2 Steenrod algebra, serre-cartan basis
sage: SteenrodAlgebra(profile=(3,2,1,0)) sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] sage: SteenrodAlgebra(profile=lambda n: 4) sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [4, 4, 4, ..., 4, 4, +Infinity, +Infinity, +Infinity, ...] sage: SteenrodAlgebra(p=5, profile=(lambda n: 4, lambda n: 1)) sub-Hopf algebra of mod 5 Steenrod algebra, milnor basis, profile function ([4, 4, 4, ..., 4, 4, +Infinity, +Infinity, +Infinity, ...], [1, 1, 1, ..., 1, 1, 2, 2, ...]) """ """ String rep for list ``l`` if ``l`` is short enough; otherwise print the first few terms and the last few terms, with an ellipsis in between. """ else:
else:
def _latex_(self): r""" LaTeX representation of the Steenrod algebra.
EXAMPLES::
sage: C = SteenrodAlgebra(3) sage: C mod 3 Steenrod algebra, milnor basis sage: C._latex_() '\\mathcal{A}_{3}' """
def _repr_term(self, t): r""" String representation of the monomial specified by the tuple ``t``.
INPUT:
- ``t`` - tuple, representing basis element in the current basis.
OUTPUT: string
This is tested in many places: any place elements are printed is essentially a doctest for this method. Also, each basis has its own method for printing monomials, and those are doctested individually. We give a few doctests here, in addition.
EXAMPLES::
sage: SteenrodAlgebra()._repr_term((3,2)) 'Sq(3,2)' sage: SteenrodAlgebra(p=7)._repr_term(((0,2), (3,2))) 'Q_0 Q_2 P(3,2)' sage: SteenrodAlgebra(basis='adem')._repr_term((14,2)) 'Sq^14 Sq^2' sage: SteenrodAlgebra(basis='adem', p=3)._repr_term((1,3,0)) 'beta P^3' sage: SteenrodAlgebra(basis='pst')._repr_term(((0,2), (1,3))) 'P^0_2 P^1_3' sage: SteenrodAlgebra(basis='arnon_a')._repr_term(((0,2), (1,3))) 'X^0_2 X^1_3'
sage: A7 = SteenrodAlgebra(7) sage: x = A7.Q(0,3) * A7.P(2,2) sage: x._repr_() 'Q_0 Q_3 P(2,2)' sage: x Q_0 Q_3 P(2,2) sage: a = SteenrodAlgebra().Sq(0,0,2) sage: a Sq(0,0,2) sage: A2_adem = SteenrodAlgebra(2,'admissible') sage: A2_adem(a) Sq^8 Sq^4 Sq^2 + Sq^9 Sq^4 Sq^1 + Sq^10 Sq^3 Sq^1 + Sq^10 Sq^4 + Sq^11 Sq^2 Sq^1 + Sq^12 Sq^2 + Sq^13 Sq^1 + Sq^14 sage: SteenrodAlgebra(2, 'woodz')(a) Sq^6 Sq^7 Sq^1 + Sq^14 + Sq^4 Sq^7 Sq^3 + Sq^4 Sq^7 Sq^2 Sq^1 + Sq^12 Sq^2 + Sq^8 Sq^6 + Sq^8 Sq^4 Sq^2 sage: SteenrodAlgebra(2, 'arnonc')(a) Sq^4 Sq^2 Sq^8 + Sq^4 Sq^4 Sq^6 + Sq^4 Sq^6 Sq^4 + Sq^6 Sq^8 + Sq^8 Sq^4 Sq^2 + Sq^8 Sq^6 sage: SteenrodAlgebra(2, 'pst_llex')(a) P^1_3 sage: SteenrodAlgebra(2, 'comm_revz')(a) c_0,1 c_1,1 c_0,3 c_2,1 + c_0,2 c_0,3 c_2,1 + c_1,3 sage: SteenrodAlgebra(2, generic=True, basis='pst').P(0,0,2) P^1_3 """ serre_cartan_mono_to_string, wood_mono_to_string, \ wall_mono_to_string, wall_long_mono_to_string, \ arnonA_mono_to_string, arnonA_long_mono_to_string, \ pst_mono_to_string, \ comm_long_mono_to_string, comm_mono_to_string
def _latex_term(self, t): """ LaTeX representation of the monomial specified by the tuple ``t``.
INPUT:
- ``t`` - tuple, representing basis element in the current basis.
OUTPUT: string
The string depends on the basis over which the element is defined.
EXAMPLES::
sage: A7 = SteenrodAlgebra(7) sage: A7._latex_term(((0, 3), (2,2))) 'Q_{0} Q_{3} \\mathcal{P}(2,2)' sage: x = A7.Q(0,3) * A7.P(2,2) sage: x._latex_() # indirect doctest 'Q_{0} Q_{3} \\mathcal{P}(2,2)' sage: latex(x) Q_{0} Q_{3} \mathcal{P}(2,2) sage: b = Sq(0,2) sage: b.change_basis('adem')._latex_() '\\text{Sq}^{4} \\text{Sq}^{2} + \\text{Sq}^{5} \\text{Sq}^{1} + \\text{Sq}^{6}' sage: b.change_basis('woody')._latex_() '\\text{Sq}^{2} \\text{Sq}^{3} \\text{Sq}^{1} + \\text{Sq}^{6} + \\text{Sq}^{4} \\text{Sq}^{2}' sage: SteenrodAlgebra(2, 'arnona')(b)._latex_() 'X^{1}_{1} X^{2}_{2} + X^{2}_{1}' sage: SteenrodAlgebra(p=3, basis='serre-cartan').Q(0)._latex_() '\\beta' sage: latex(Sq(2).change_basis('adem').coproduct()) 1 \otimes \text{Sq}^{2} + \text{Sq}^{1} \otimes \text{Sq}^{1} + \text{Sq}^{2} \otimes 1 sage: latex(SteenrodAlgebra(basis='pst').P(0,0,2)) P^{1}_{3} """
def __eq__(self, right): r""" Two Steenrod algebras are equal iff their associated primes, bases, and profile functions (if present) are equal. Because this class inherits from :class:`UniqueRepresentation`, this means that they are equal if and only they are identical: ``A == B`` is True if and only if ``A is B`` is True.
EXAMPLES::
sage: A = SteenrodAlgebra(2) sage: B = SteenrodAlgebra(2, 'adem') sage: A == B False sage: C = SteenrodAlgebra(17) sage: A == C False
sage: A1 = SteenrodAlgebra(2, profile=[2,1]) sage: A1 == A False sage: A1 == SteenrodAlgebra(2, profile=[2,1,0]) True sage: A1 == SteenrodAlgebra(2, profile=[2,1], basis='pst') False """
def __ne__(self, right): r""" The negation of the method ``__eq__``.
EXAMPLES::
sage: SteenrodAlgebra(p=2) != SteenrodAlgebra(p=2, profile=[2,1]) True """
def profile(self, i, component=0): r""" Profile function for this algebra.
INPUT:
- `i` - integer - ``component`` - either 0 or 1, optional (default 0)
OUTPUT: integer or `\infty`
See the documentation for :mod:`sage.algebras.steenrod.steenrod_algebra` and :func:`SteenrodAlgebra` for information on profile functions.
This applies the profile function to the integer `i`. Thus when `p=2`, `i` must be a positive integer. When `p` is odd, there are two profile functions, `e` and `k` (in the notation of the aforementioned documentation), corresponding, respectively to ``component=0`` and ``component=1``. So when `p` is odd and ``component`` is 0, `i` must be positive, while when ``component`` is 1, `i` must be non-negative.
EXAMPLES::
sage: SteenrodAlgebra().profile(3) +Infinity sage: SteenrodAlgebra(profile=[3,2,1]).profile(1) 3 sage: SteenrodAlgebra(profile=[3,2,1]).profile(2) 2
When the profile is specified by a list, the default behavior is to return zero values outside the range of the list. This can be overridden if the algebra is created with an infinite ``truncation_type``::
sage: SteenrodAlgebra(profile=[3,2,1]).profile(9) 0 sage: SteenrodAlgebra(profile=[3,2,1], truncation_type=Infinity).profile(9) +Infinity
sage: B = SteenrodAlgebra(p=3, profile=(lambda n: n, lambda n: 1)) sage: B.profile(3) 3 sage: B.profile(3, component=1) 1
sage: EA = SteenrodAlgebra(generic=True, profile=(lambda n: n, lambda n: 1)) sage: EA.profile(4) 4 sage: EA.profile(2, component=1) 1 """ # determine the tuple t to use else: # case 1: exponents of the xi's return 0 else: # case 2: exponents of the tau's return 1 else:
def homogeneous_component(self, n): """ Return the nth homogeneous piece of the Steenrod algebra.
INPUT:
- `n` - integer
OUTPUT: a vector space spanned by the basis for this algebra in dimension `n`
EXAMPLES::
sage: A = SteenrodAlgebra() sage: A.homogeneous_component(4) Vector space spanned by (Sq(1,1), Sq(4)) over Finite Field of size 2 sage: SteenrodAlgebra(profile=[2,1,0]).homogeneous_component(4) Vector space spanned by (Sq(1,1),) over Finite Field of size 2
The notation A[n] may also be used::
sage: A[5] Vector space spanned by (Sq(2,1), Sq(5)) over Finite Field of size 2 sage: SteenrodAlgebra(basis='wall')[4] Vector space spanned by (Q^1_0 Q^0_0, Q^2_2) over Finite Field of size 2 sage: SteenrodAlgebra(p=5)[17] Vector space spanned by (Q_1 P(1), Q_0 P(2)) over Finite Field of size 5
Note that A[n] is just a vector space, not a Hopf algebra, so its elements don't have products, coproducts, or antipodes defined on them. If you want to use operations like this on elements of some A[n], then convert them back to elements of A::
sage: A[5].basis() Finite family {(5,): milnor[(5,)], (2, 1): milnor[(2, 1)]} sage: a = list(A[5].basis())[1] sage: a # not in A, doesn't print like an element of A milnor[(5,)] sage: A(a) # in A Sq(5) sage: A(a) * A(a) Sq(7,1) sage: a * A(a) # only need to convert one factor Sq(7,1) sage: a.antipode() # not defined Traceback (most recent call last): ... AttributeError: 'CombinatorialFreeModule_with_category.element_class' object has no attribute 'antipode' sage: A(a).antipode() # convert to elt of A, then compute antipode Sq(2,1) + Sq(5)
sage: G = SteenrodAlgebra(p=5, profile=[[2,1], [2,2,2]], basis='pst')
TESTS:
The following sort of thing is also tested by the function :func:`steenrod_basis_error_check <sage.algebras.steenrod.steenrod_algebra_bases.steenrod_basis_error_check>`::
sage: H = SteenrodAlgebra(p=5, profile=[[2,1], [2,2,2]]) sage: G = SteenrodAlgebra(p=5, profile=[[2,1], [2,2,2]], basis='pst') sage: max([H[n].dimension() - G[n].dimension() for n in range(100)]) 0 """ element_class=self.Element, prefix=self._basis_name)
__getitem__ = homogeneous_component
def one_basis(self): """ The index of the element 1 in the basis for the Steenrod algebra.
EXAMPLES::
sage: SteenrodAlgebra(p=2).one_basis() () sage: SteenrodAlgebra(p=7).one_basis() ((), ()) """
def product_on_basis(self, t1, t2): """ The product of two basis elements of this algebra
INPUT:
- ``t1``, ``t2`` -- tuples, the indices of two basis elements of self
OUTPUT: the product of the two corresponding basis elements, as an element of self
ALGORITHM: If the two elements are represented in the Milnor basis, use Milnor multiplication as implemented in :mod:`sage.algebras.steenrod.steenrod_algebra_mult`. If the two elements are represented in the Serre-Cartan basis, then multiply them using Adem relations (also implemented in :mod:`sage.algebras.steenrod.steenrod_algebra_mult`). This provides a good way of checking work -- multiply Milnor elements, then convert them to Adem elements and multiply those, and see if the answers correspond.
If the two elements are represented in some other basis, then convert them both to the Milnor basis and multiply.
EXAMPLES::
sage: Milnor = SteenrodAlgebra() sage: Milnor.product_on_basis((2,), (2,)) Sq(1,1) sage: Adem = SteenrodAlgebra(basis='adem') sage: Adem.Sq(2) * Adem.Sq(2) # indirect doctest Sq^3 Sq^1
When multiplying elements from different bases, the left-hand factor determines the form of the output::
sage: Adem.Sq(2) * Milnor.Sq(2) Sq^3 Sq^1 sage: Milnor.Sq(2) * Adem.Sq(2) Sq(1,1)
TESTS::
sage: all([Adem(Milnor.Sq(n) ** 3)._repr_() == (Adem.Sq(n) ** 3)._repr_() for n in range(10)]) True sage: Wall = SteenrodAlgebra(basis='wall') sage: Wall(Adem.Sq(4,4) * Milnor.Sq(4)) == Adem(Wall.Sq(4,4) * Milnor.Sq(4)) True
sage: A3 = SteenrodAlgebra(p=3, basis='adem') sage: M3 = SteenrodAlgebra(p=3, basis='milnor') sage: all([A3(M3.P(n) * M3.Q(0) * M3.P(n))._repr_() == (A3.P(n) * A3.Q(0) * A3.P(n))._repr_() for n in range(5)]) True
sage: EA = SteenrodAlgebra(generic=True) sage: EA.product_on_basis(((1, 3), (2, 1)), ((2, ), (0, 0, 1))) Q_1 Q_2 Q_3 P(2,1,1)
sage: EA2 = SteenrodAlgebra(basis='serre-cartan', generic=True) sage: EA2.product_on_basis((1, 2, 0, 1, 0), (1, 2, 0, 1, 0)) beta P^4 P^2 beta + beta P^5 beta P^1 """ else: # make sure output has an odd number of terms. if both t1 # and t2 have an odd number, concatenate them, adding the # middle term... # # if either t1 or t2 has an even number of terms, append a # 0. t1 = t1 + (0,) t2 = t2 + (0,) else: # p=2 else:
def coproduct_on_basis(self, t, algorithm=None): r""" The coproduct of a basis element of this algebra
INPUT:
- ``t`` -- tuple, the index of a basis element of self
- ``algorithm`` -- ``None`` or a string, either 'milnor' or 'serre-cartan' (or anything which will be converted to one of these by the function :func:`get_basis_name <sage.algebras.steenrod.steenrod_algebra_misc.get_basis_name>`. If ``None``, default to 'milnor' unless current basis is 'serre-cartan', in which case use 'serre-cartan'.
ALGORITHM: The coproduct on a Milnor basis element `P(n_1, n_2, ...)` is `\sum P(i_1, i_2, ...) \otimes P(j_1, j_2, ...)`, summed over all `i_k + j_k = n_k` for each `k`. At odd primes, each element `Q_n` is primitive: its coproduct is `Q_n \otimes 1 + 1 \otimes Q_n`.
One can deduce a coproduct formula for the Serre-Cartan basis from this: the coproduct on each `P^n` is `\sum P^i \otimes P^{n-i}` and at odd primes `\beta` is primitive. Since the coproduct is an algebra map, one can then compute the coproduct on any Serre-Cartan basis element.
Which of these methods is used is controlled by whether ``algorithm`` is 'milnor' or 'serre-cartan'.
OUTPUT: the coproduct of the corresponding basis element, as an element of ``self`` tensor ``self``.
EXAMPLES::
sage: A = SteenrodAlgebra() sage: A.coproduct_on_basis((3,)) 1 # Sq(3) + Sq(1) # Sq(2) + Sq(2) # Sq(1) + Sq(3) # 1
TESTS::
sage: all([A.coproduct_on_basis((n,1), algorithm='milnor') == A.coproduct_on_basis((n,1), algorithm='adem') for n in range(9)]) # long time True sage: A7 = SteenrodAlgebra(p=7, basis='adem') sage: all([A7.coproduct_on_basis((0,n,1), algorithm='milnor') == A7.coproduct_on_basis((0,n,1), algorithm='adem') for n in range(9)]) # long time True """ """ if t = (n0, n1, ...), then return list of terms (i0, i1, ...) where ik <= nk for each k. From each such term, can recover the second factor in the coproduct. """
algorithm = 'serre-cartan' else: else: # trim trailing zeros: else: # p odd # trim trailing zeros: else: else: else: else: A._change_basis_on_basis(b, basis))),coeff))
def coproduct(self, x, algorithm='milnor'): r""" Return the coproduct of an element ``x`` of this algebra.
INPUT:
- ``x`` -- element of self
- ``algorithm`` -- ``None`` or a string, either 'milnor' or 'serre-cartan' (or anything which will be converted to one of these by the function :func:`get_basis_name <sage.algebras.steenrod.steenrod_algebra_misc.get_basis_name>`. If ``None``, default to 'serre-cartan' if current basis is 'serre-cartan'; otherwise use 'milnor'.
This calls :meth:`coproduct_on_basis` on the summands of ``x`` and extends linearly.
EXAMPLES::
sage: SteenrodAlgebra().Sq(3).coproduct() 1 # Sq(3) + Sq(1) # Sq(2) + Sq(2) # Sq(1) + Sq(3) # 1
The element `\text{Sq}(0,1)` is primitive::
sage: SteenrodAlgebra(basis='adem').Sq(0,1).coproduct() 1 # Sq^2 Sq^1 + 1 # Sq^3 + Sq^2 Sq^1 # 1 + Sq^3 # 1 sage: SteenrodAlgebra(basis='pst').Sq(0,1).coproduct() 1 # P^0_2 + P^0_2 # 1
sage: SteenrodAlgebra(p=3).P(4).coproduct() 1 # P(4) + P(1) # P(3) + P(2) # P(2) + P(3) # P(1) + P(4) # 1 sage: SteenrodAlgebra(p=3).P(4).coproduct(algorithm='serre-cartan') 1 # P(4) + P(1) # P(3) + P(2) # P(2) + P(3) # P(1) + P(4) # 1 sage: SteenrodAlgebra(p=3, basis='serre-cartan').P(4).coproduct() 1 # P^4 + P^1 # P^3 + P^2 # P^2 + P^3 # P^1 + P^4 # 1 sage: SteenrodAlgebra(p=11, profile=((), (2,1,2))).Q(0,2).coproduct() 1 # Q_0 Q_2 + Q_0 # Q_2 + Q_0 Q_2 # 1 + 10*Q_2 # Q_0 """ # taken from categories.coalgebras_with_basis, then modified # to allow the use of the "algorithm" keyword
def antipode_on_basis(self, t): r""" The antipode of a basis element of this algebra
INPUT:
- ``t`` -- tuple, the index of a basis element of self
OUTPUT: the antipode of the corresponding basis element, as an element of self.
ALGORITHM: according to a result of Milnor's, the antipode of `\text{Sq}(n)` is the sum of all of the Milnor basis elements in dimension `n`. So: convert the element to the Serre-Cartan basis, thus writing it as a sum of products of elements `\text{Sq}(n)`, and use Milnor's formula for the antipode of `\text{Sq}(n)`, together with the fact that the antipode is an antihomomorphism: if we call the antipode `c`, then `c(ab) = c(b) c(a)`.
At odd primes, a similar method is used: the antipode of `P(n)` is the sum of the Milnor P basis elements in dimension `n*2(p-1)`, multiplied by `(-1)^n`, and the antipode of `\beta = Q_0` is `-Q_0`. So convert to the Serre-Cartan basis, as in the `p=2` case.
EXAMPLES::
sage: A = SteenrodAlgebra() sage: A.antipode_on_basis((4,)) Sq(1,1) + Sq(4) sage: A.Sq(4).antipode() Sq(1,1) + Sq(4) sage: Adem = SteenrodAlgebra(basis='adem') sage: Adem.Sq(4).antipode() Sq^3 Sq^1 + Sq^4 sage: SteenrodAlgebra(basis='pst').Sq(3).antipode() P^0_1 P^1_1 + P^0_2 sage: a = SteenrodAlgebra(basis='wall_long').Sq(10) sage: a.antipode() Sq^1 Sq^2 Sq^4 Sq^1 Sq^2 + Sq^2 Sq^4 Sq^1 Sq^2 Sq^1 + Sq^8 Sq^2 sage: a.antipode().antipode() == a True
sage: SteenrodAlgebra(p=3).P(6).antipode() P(2,1) + P(6) sage: SteenrodAlgebra(p=3).P(6).antipode().antipode() P(6)
TESTS::
sage: Milnor = SteenrodAlgebra() sage: all([x.antipode().antipode() == x for x in Milnor.basis(11)]) # long time True sage: A5 = SteenrodAlgebra(p=5, basis='adem') sage: all([x.antipode().antipode() == x for x in A5.basis(25)]) True sage: H = SteenrodAlgebra(profile=[2,2,1]) sage: H.Sq(1,2).antipode() in H True """ else: else:
def counit_on_basis(self, t): """ The counit sends all elements of positive degree to zero.
INPUT:
- ``t`` -- tuple, the index of a basis element of self
EXAMPLES::
sage: A2 = SteenrodAlgebra(p=2) sage: A2.counit_on_basis(()) 1 sage: A2.counit_on_basis((0,0,1)) 0 sage: parent(A2.counit_on_basis((0,0,1))) Finite Field of size 2 sage: A3 = SteenrodAlgebra(p=3) sage: A3.counit_on_basis(((1,2,3), (1,1,1))) 0 sage: A3.counit_on_basis(((), ())) 1 sage: A3.counit(A3.P(10,5)) 0 sage: A3.counit(A3.P(0)) 1 """ else:
def _milnor_on_basis(self, t): r""" Convert the tuple ``t`` in the current basis to an element in the Milnor basis.
INPUT:
- ``t`` - tuple, representing basis element in the current basis.
OUTPUT: element of the Steenrod algebra with the Milnor basis
ALGORITHM: there is a simple conversion from each basis to the Milnor basis, so use that. In more detail:
- If the current basis is the Milnor basis, just return the corresponding element.
- If the current basis is the Serre-Cartan basis: when `p=2`, the element `\text{Sq}^a` equals the Milnor element `\text{Sq}(a)`; when `p` is odd, `\mathcal{P}^a = \mathcal{P}(a)` and `\beta = Q_0`. Hence for any Serre-Cartan basis element, represent it in the Milnor basis by computing an appropriate product using Milnor multiplication.
- The same goes for Arnon's C basis, since the elements are monomials in the Steenrod squares.
- If the current basis is Wood's Y or Z bases, then each basis element is a monomial in the classes `w(m,k) = \text{Sq}^{2^m (2^{k+1}-1)}`. So again, multiply the corresponding Milnor elements together.
- The Wall basis: each basis element is a monomial in the elements `Q^m_k = Sq(2^k) Sq(2^{k+1}) ... Sq(2^m)`.
- Arnon's A basis: each basis element is a monomial in the elements `X^m_k = Sq(2^m) ... Sq(2^{k+1}) Sq(2^k)`.
- The `P^s_t` bases: when `p=2`, each basis element is a monomial in the elements `P^s_t`. When `p` is odd, each basis element is a product of elements `Q_i` and a monomial in the elements `(P^s_t)^n` where `0 < n < p`.
- The commutator bases: when `p=2`, each basis element is a monomial in the iterated commutators `c_{i,j}`, defined by `c_{i,1} = \text{Sq}(2^i)` and `c_{i,j} = [c_{i,j-1}, \text{Sq}(2^{i+j-1})]`. When `p` is odd, each basis element is a product of elements `Q_i` and a monomial in the elements `c_{i,j}^n` where `0 < n < p`, `c_{i,1} = P(p^i)` and `c_{i,j} = [P(p^{i+j-1}), c_{i,j-1}]`.
EXAMPLES::
sage: Adem = SteenrodAlgebra(basis='serre-cartan') sage: Adem._milnor_on_basis((2,1)) # Sq^2 Sq^1 Sq(0,1) + Sq(3) sage: Pst = SteenrodAlgebra(basis='pst') sage: Pst._milnor_on_basis(((0,1), (1,1), (2,1))) Sq(7) """ # milnor
# serre-cartan, arnonc
else: # wood_y: # each entry in t is a pair (m,k), corresponding to w(m,k), defined by # `w(m,k) = \text{Sq}^{2^m (2^{k+1}-1)}`.
# wall[_long] # each entry in t is a pair (m,k), corresponding to Q^m_k, defined by #`Q^m_k = Sq(2^k) Sq(2^{k+1}) ... Sq(2^m)`.
# pst... # each entry in t is a pair (i,j), corresponding to P^i_j else: # t = (Q, P) where Q is the tuple of Q_i's, and P is a # tuple with entries of the form ((i,j), n), # corresponding to (P^i_j)^n
# arnona[_long] # each entry in t is a pair (m,k), corresponding to X^m_k, defined by # `X^m_k = Sq(2^m) ... Sq(2^{k+1}) Sq(2^k)`
# comm...[_long] # each entry in t is a pair (i,j), corresponding to # c_{i,j}, the iterated commutator defined by c_{i,1} # = Sq(2^i) and c_{i,j} = [c_{i,j-1}, Sq(2^{i+j-1})]. else: # t = (Q, P) where Q is the tuple of Q_i's, and P is a # tuple with entries of the form ((i,j), n), # corresponding to (c_{i,j})^n. Here c_{i,j} is the # iterated commutator defined by c_{i,1} = P(p^i) and # c_{i,j} = [P(p^{i+j-1}), c_{i,j-1}]. if len(t[0]) > 0: ans = ans * A.Q(*t[0]) for ((i, j), n) in t[1]: comm = A.P(p**i) for k in range(2, j+1): y = A.P(p**(i+k-1)) comm = y * comm - comm * y ans = ans * comm**n
@lazy_attribute def milnor(self): """ Convert an element of this algebra to the Milnor basis
INPUT:
- x -- an element of this algebra
OUTPUT: x converted to the Milnor basis
ALGORITHM: use the method ``_milnor_on_basis`` and linearity.
EXAMPLES::
sage: Adem = SteenrodAlgebra(basis='adem') sage: a = Adem.Sq(2) * Adem.Sq(1) sage: Adem.milnor(a) Sq(0,1) + Sq(3) """
def _change_basis_on_basis(self, t, basis='milnor'): """ Convert the tuple t to the named basis.
INPUT:
- ``t`` - tuple, representing basis element in the current basis.
- ``basis`` - string, the basis to which to convert, optional (default 'milnor')
OUTPUT: an element of the Steenrod algebra with basis ``basis``.
ALGORITHM: it's straightforward to convert to the Milnor basis (using :meth:`milnor` or :meth:`_milnor_on_basis`), so it's straightforward to produce a matrix representing this conversion in any degree. The function :func:`convert_from_milnor_matrix <steenrod_algebra_bases.convert_from_milnor_matrix>` provides the inverse operation.
So: convert from the current basis to the Milnor basis, then from the Milnor basis to the new basis.
EXAMPLES::
sage: Adem = SteenrodAlgebra(basis='adem') sage: a = Adem({(2,1): 1}); a Sq^2 Sq^1 sage: a.change_basis('adem') # indirect doctest Sq^2 Sq^1 sage: a.change_basis('milnor') Sq(0,1) + Sq(3) sage: a.change_basis('pst') P^0_1 P^1_1 + P^0_2 sage: a.change_basis('milnor').change_basis('adem').change_basis('adem') Sq^2 Sq^1 sage: a.change_basis('wall') == a.change_basis('woody') True
TESTS::
sage: a = sum(SteenrodAlgebra(basis='comm').basis(10)) sage: a.change_basis('adem').change_basis('wall').change_basis('comm')._repr_() == a._repr_() True sage: a.change_basis('pst').change_basis('milnor').change_basis('comm')._repr_() == a._repr_() True sage: a.change_basis('woody').change_basis('arnona').change_basis('comm')._repr_() == a._repr_() True
sage: b = sum(SteenrodAlgebra(p=3).basis(41)) sage: b.change_basis('adem').change_basis('adem').change_basis('milnor')._repr_() == b._repr_() True
sage: SteenrodAlgebra(generic=True).P(0,2).change_basis('serre-cartan') P^4 P^2 + P^5 P^1 + P^6 """ convert_from_milnor_matrix convert_from_milnor_matrix(deg, basis, p, generic=self._generic))
def _change_basis(self, x, basis='milnor'): """ Convert an element of this algebra to the specified basis
INPUT:
- ``x`` - an element of this algebra.
- ``basis`` - string, the basis to which to convert, optional (default 'milnor')
OUTPUT: an element of the Steenrod algebra with basis ``basis``.
ALGORITHM: use :meth:`_change_basis_on_basis` and linearity
EXAMPLES::
sage: Adem = SteenrodAlgebra(basis='adem') sage: a = Adem({(2,1): 1}); a Sq^2 Sq^1 sage: a.change_basis('adem') # indirect doctest Sq^2 Sq^1 sage: a.change_basis('milnor') Sq(0,1) + Sq(3) sage: a.change_basis('pst') P^0_1 P^1_1 + P^0_2 """
def degree_on_basis(self, t): r""" The degree of the monomial specified by the tuple ``t``.
INPUT:
- ``t`` - tuple, representing basis element in the current basis.
OUTPUT: integer, the degree of the corresponding element.
The degree of `\text{Sq}(i_1,i_2,i_3,...)` is
.. MATH::
i_1 + 3i_2 + 7i_3 + ... + (2^k - 1) i_k + ....
At an odd prime `p`, the degree of `Q_k` is `2p^k - 1` and the degree of `\mathcal{P}(i_1, i_2, ...)` is
.. MATH::
\sum_{k \geq 0} 2(p^k - 1) i_k.
ALGORITHM: Each basis element is represented in terms relevant to the particular basis: 'milnor' basis elements (at the prime 2) are given by tuples ``(a,b,c,...)`` corresponding to the element `\text{Sq}(a,b,c,...)`, while 'pst' basis elements are given by tuples of pairs ``((a, b), (c, d), ...)``, corresponding to the product `P^a_b P^c_d ...`. The other bases have similar descriptions. The degree of each basis element is computed from this data, rather than converting the element to the Milnor basis, for example, and then computing the degree.
EXAMPLES::
sage: SteenrodAlgebra().degree_on_basis((0,0,1)) 7 sage: Sq(7).degree() 7
sage: A11 = SteenrodAlgebra(p=11) sage: A11.degree_on_basis(((), (1,1))) 260 sage: A11.degree_on_basis(((2,), ())) 241 """ """ For m=(n_1, n_2, n_3, ...), Sum_i (mult) * n_i * (p^i - 1) """
""" For m=(n_0, n_1, n_2, ...), Sum_i 2*p^(n_i) - 1 """
# milnor else: # serre-cartan, arnonc n += 1 else:
# wood_y: # each entry in t is a pair (m,k), corresponding to w(m,k), defined by # `w(m,k) = \text{Sq}^{2^m (2^{k+1}-1)}`.
# wall, arnon_a # Wall: each entry in t is a pair (m,k), corresponding to # Q^m_k, defined by `Q^m_k = Sq(2^k) Sq(2^{k+1}) # ... Sq(2^m)`. # # Arnon A: each entry in t is a pair (m,k), corresponding # to X^m_k, defined by `X^m_k = Sq(2^m) ... Sq(2^{k+1}) # Sq(2^k)`
# pst, comm # Pst: each entry in t is a pair (i,j), corresponding to P^i_j # # Comm: each entry in t is a pair (i,j), corresponding # to c_{i,j}, the iterated commutator defined by # c_{i,1} = Sq(2^i) and c_{i,j} = [c_{i,j-1}, # Sq(2^{i+j-1})]. # p odd: # # Pst: have pair (Q, P) where Q is a tuple of Q's, as in # the Milnor basis, and P is a tuple of terms of the form # ((i,j), n), corresponding to (P^i_j)^n. # # Comm: similarly (Q, C) with Q as above and C a tuple # with each entry in t is of the form ((s,t), n), # corresponding to c_{s,t}^n. here c_{s,t} is the # iterated commutator defined by c_{s,1} = P(p^s) and # c_{s,t} = [P(p^{s+t-1}), c_{s,t-1}]. q_deg = q_degree(t[0], prime=p) p_deg = sum(2 * n * p**s * (p**t - 1) for ((s,t), n) in t[1]) return q_deg + p_deg
# coercion methods:
def _coerce_map_from_(self, S): r""" True if there is a coercion from ``S`` to ``self``, False otherwise.
INPUT:
- ``S`` - a Sage object.
The algebras that coerce into the mod p Steenrod algebra are:
- the mod p Steenrod algebra `A` - its sub-Hopf algebras - its homogeneous components - its base field `GF(p)` - `ZZ`
Similarly, a sub-Hopf algebra `B` of `A` coerces into another sub-Hopf algebra `C` if and only if the profile function for `B` is less than or equal to that of `C`, pointwise.
EXAMPLES::
sage: A = SteenrodAlgebra() sage: A1 = SteenrodAlgebra(profile=[2,1]) sage: A2 = SteenrodAlgebra(profile=[3,2,1]) sage: B = SteenrodAlgebra(profile=[1,2,1]) sage: A._coerce_map_from_(A1) True sage: A2._coerce_map_from_(A1) True sage: A1._coerce_map_from_(A) False sage: A1._coerce_map_from_(B) False sage: B._coerce_map_from_(A1) False
sage: A._coerce_map_from_(A[12]) True
sage: EA = SteenrodAlgebra(generic=True) sage: A._coerce_map_from_(EA) False sage: EA._coerce_map_from_(A) False sage: EA._coerce_map_from_(EA) True
sage: A3 = SteenrodAlgebra(p=3) sage: A31 = SteenrodAlgebra(p=3, profile=([1], [2, 2])) sage: B3 = SteenrodAlgebra(p=3, profile=([1, 2, 1], [1])) sage: A3._coerce_map_from_(A31) True sage: A31._coerce_map_from_(A3) False sage: A31._coerce_map_from_(B3) False sage: B3._coerce_map_from_(A31) False """ # deal with profiles. for i in range(1, max(self_prec, S_prec)+1)]) for i in range(1, max(self_prec, S_prec)+1)]) and all([self.profile(i, 1) >= S.profile(i, 1) for i in range(1, max(self_prec, S_prec)+1)])) and S.dimension() < Infinity and p == S.base_ring().characteristic()): # return all([a in self for a in S.basis()]) except ValueError: return False
def _element_constructor_(self, x): r""" Try to turn ``x`` into an element of ``self``.
INPUT:
- ``x`` - an element of some Steenrod algebra or an element of `\ZZ` or `\GF{p}` or a dict
OUTPUT: ``x`` as a member of ``self``.
If ``x`` is a dict, then call :meth:`_from_dict` on it, coercing the coefficients into the base field. That is, treat it as having entries of the form ``tuple: coeff``, where ``tuple`` is a tuple representing a basis element and ``coeff`` is the coefficient of that element.
EXAMPLES::
sage: A1 = SteenrodAlgebra(profile=[2,1]) sage: A1(Sq(2)) # indirect doctest Sq(2) sage: A1._element_constructor_(Sq(2)) Sq(2) sage: A1(3) # map integer into A1 1 sage: A1._element_constructor_(Sq(4)) # Sq(4) not in A1 Traceback (most recent call last): ... ValueError: Element does not lie in this Steenrod algebra sage: A1({(2,): 1, (1,): 13}) Sq(1) + Sq(2) """ return self.from_base_ring_from_one_basis(x)
else:
def __contains__(self, x): r""" True if self contains x.
EXAMPLES::
sage: Sq(3,1,1) in SteenrodAlgebra() True sage: Sq(3,1,1) in SteenrodAlgebra(p=5) False
sage: A1 = SteenrodAlgebra(profile=[2,1]) sage: Sq(3) in A1 True sage: Sq(4) in A1 False sage: Sq(0,2) in A1 False
sage: Sq(3) in SteenrodAlgebra(generic=True) False
sage: A_3 = SteenrodAlgebra(p=3) sage: B_3 = SteenrodAlgebra(p=3, profile=([1], [2,2,1,1])) sage: A_3.P(2) in B_3 True sage: A_3.P(3) in B_3 False sage: A_3.Q(1) in B_3 True sage: A_3.P(1) * A_3.Q(2) in B_3 False """ return True and x.prime() == p): for mono in A(x).support()])
def basis(self, d=None): """ Returns basis for self, either the whole basis or the basis in degree `d`.
INPUT:
- `d` -- integer or ``None``, optional (default ``None``)
OUTPUT: If `d` is ``None``, then return a basis of the algebra. Otherwise, return the basis in degree `d`.
EXAMPLES::
sage: A3 = SteenrodAlgebra(3) sage: A3.basis(13) Family (Q_1 P(2), Q_0 P(3)) sage: SteenrodAlgebra(2, 'adem').basis(12) Family (Sq^12, Sq^11 Sq^1, Sq^9 Sq^2 Sq^1, Sq^8 Sq^3 Sq^1, Sq^10 Sq^2, Sq^9 Sq^3, Sq^8 Sq^4)
sage: A = SteenrodAlgebra(profile=[1,2,1]) sage: A.basis(2) Family () sage: A.basis(3) Family (Sq(0,1),) sage: SteenrodAlgebra().basis(3) Family (Sq(0,1), Sq(3)) sage: A_pst = SteenrodAlgebra(profile=[1,2,1], basis='pst') sage: A_pst.basis(3) Family (P^0_2,)
sage: A7 = SteenrodAlgebra(p=7) sage: B = SteenrodAlgebra(p=7, profile=([1,2,1], [1])) sage: A7.basis(84) Family (P(7),) sage: B.basis(84) Family () sage: C = SteenrodAlgebra(p=7, profile=([1], [2,2])) sage: A7.Q(0,1) in C.basis(14) True sage: A7.Q(2) in A7.basis(97) True sage: A7.Q(2) in C.basis(97) False
With no arguments, return the basis of the whole algebra. This does not print in a very helpful way, unfortunately::
sage: A7.basis() Lazy family (Term map from basis key family of mod 7 Steenrod algebra, milnor basis to mod 7 Steenrod algebra, milnor basis(i))_{i in basis key family of mod 7 Steenrod algebra, milnor basis} sage: for (idx,a) in zip((1,..,9),A7.basis()): ....: print("{} {}".format(idx, a)) 1 1 2 Q_0 3 P(1) 4 Q_1 5 Q_0 P(1) 6 Q_0 Q_1 7 P(2) 8 Q_1 P(1) 9 Q_0 P(2) sage: D = SteenrodAlgebra(p=3, profile=([1], [2,2])) sage: sorted(D.basis()) [1, P(1), P(2), Q_0, Q_0 P(1), Q_0 P(2), Q_0 Q_1, Q_0 Q_1 P(1), Q_0 Q_1 P(2), Q_1, Q_1 P(1), Q_1 P(2)] """ else:
def _check_profile_on_basis(self, t): """ True if the element specified by the tuple ``t`` is in this algebra.
INPUT:
- ``t`` - tuple of ...
EXAMPLES::
sage: A = SteenrodAlgebra(profile=[1,2,1]) sage: A._check_profile_on_basis((0,0,1)) True sage: A._check_profile_on_basis((0,0,2)) False sage: A5 = SteenrodAlgebra(p=5, profile=([3,2,1], [2,2,2,2,2])) sage: A5._check_profile_on_basis(((), (1,5))) True sage: A5._check_profile_on_basis(((1,1,1), (1,5))) True sage: A5._check_profile_on_basis(((1,1,1), (1,5,5))) False """ profile=self._profile, truncation_type=self._truncation_type, generic=self._generic) for a in self._milnor_on_basis(t)])
or t[i] < 2**self.profile(i+1) for i in range(len(t))]) # p odd: or t[1][i] < p**self.profile(i+1,0) for i in range(len(t[1]))])
def P(self, *nums): r""" The element `P(a, b, c, ...)`
INPUT:
- ``a, b, c, ...`` - non-negative integers
OUTPUT: element of the Steenrod algebra given by the Milnor single basis element `P(a, b, c, ...)`
Note that at the prime 2, this is the same element as `\text{Sq}(a, b, c, ...)`.
EXAMPLES::
sage: A = SteenrodAlgebra(2) sage: A.P(5) Sq(5) sage: B = SteenrodAlgebra(3) sage: B.P(5,1,1) P(5,1,1) sage: B.P(1,1,-12,1) Traceback (most recent call last): ... TypeError: entries must be non-negative integers
sage: SteenrodAlgebra(basis='serre-cartan').P(0,1) Sq^2 Sq^1 + Sq^3 sage: SteenrodAlgebra(generic=True).P(2,0,1) P(2,0,1) """
else: raise ValueError("Element not in this algebra")
def Q_exp(self, *nums): r""" The element `Q_0^{e_0} Q_1^{e_1} ...` , given by specifying the exponents.
INPUT:
- ``e0, e1, ...`` - sequence of 0s and 1s
OUTPUT: The element `Q_0^{e_0} Q_1^{e_1} ...`
Note that at the prime 2, `Q_n` is the element `\text{Sq}(0,0,...,1)` , where the 1 is in the `(n+1)^{st}` position.
Compare this to the method :meth:`Q`, which defines a similar element, but by specifying the tuple of subscripts of terms with exponent 1.
EXAMPLES::
sage: A2 = SteenrodAlgebra(2) sage: A5 = SteenrodAlgebra(5) sage: A2.Q_exp(0,0,1,1,0) Sq(0,0,1,1) sage: A5.Q_exp(0,0,1,1,0) Q_2 Q_3 sage: A5.Q(2,3) Q_2 Q_3 sage: A5.Q_exp(0,0,1,1,0) == A5.Q(2,3) True sage: SteenrodAlgebra(2,generic=True).Q_exp(1,0,1) Q_0 Q_2 """ raise ValueError("The tuple %s should consist " % (nums,) + \ "only of 0's and 1's") else: return self(SteenrodAlgebra(p=self.prime(),generic=self._generic).Q_exp(*nums)) else:
def Q(self, *nums): r""" The element `Q_{n0} Q_{n1} ...` , given by specifying the subscripts.
INPUT:
- ``n0, n1, ...`` - non-negative integers
OUTPUT: The element `Q_{n0} Q_{n1} ...`
Note that at the prime 2, `Q_n` is the element `\text{Sq}(0,0,...,1)` , where the 1 is in the `(n+1)^{st}` position.
Compare this to the method :meth:`Q_exp`, which defines a similar element, but by specifying the tuple of exponents.
EXAMPLES::
sage: A2 = SteenrodAlgebra(2) sage: A2.Q(2,3) Sq(0,0,1,1) sage: A5 = SteenrodAlgebra(5) sage: A5.Q(1,4) Q_1 Q_4 sage: A5.Q(1,4) == A5.Q_exp(0,1,0,0,1) True sage: H = SteenrodAlgebra(p=5, profile=[[2,1], [2,2,2]]) sage: H.Q(2) Q_2 sage: H.Q(4) Traceback (most recent call last): ... ValueError: Element not in this algebra """ return self(0) else: return self.one() else: else:
def an_element(self): """ An element of this Steenrod algebra.
The element depends on the basis and whether there is a nontrivial profile function. (This is used by the automatic test suite, so having different elements in different bases may help in discovering bugs.)
EXAMPLES::
sage: SteenrodAlgebra().an_element() Sq(2,1) sage: SteenrodAlgebra(basis='adem').an_element() Sq^4 Sq^2 Sq^1 sage: SteenrodAlgebra(p=5).an_element() 4 Q_1 Q_3 P(2,1) sage: SteenrodAlgebra(basis='pst').an_element() P^3_1 sage: SteenrodAlgebra(basis='pst', profile=[3,2,1]).an_element() P^0_1 """
else: return self.one()
return self._from_dict({((3,0),): 1, ((1, 1), (1, 0)): 1}, coerce=True) return self._from_dict({((3,3),): 1, ((1, 1), (2, 1)): 1}, coerce=True) return self._from_dict({(8,): 1, (4, 4): 1}, coerce=True) return self.term(((1,), (((1,1), 2),)), GF(p)(p-1)) if basis.find('comm') >= 0: if not self._generic: return self.monomial(((1, 2),)) return self.term(((), (((1,2), 1),)), GF(p)(p-1))
def pst(self,s,t): r""" The Margolis element `P^s_t`.
INPUT:
- ``s`` - non-negative integer
- ``t`` - positive integer
- ``p`` - positive prime number
OUTPUT: element of the Steenrod algebra
This returns the Margolis element `P^s_t` of the mod `p` Steenrod algebra: the element equal to `P(0,0,...,0,p^s)`, where the `p^s` is in position `t`.
EXAMPLES::
sage: A2 = SteenrodAlgebra(2) sage: A2.pst(3,5) Sq(0,0,0,0,8) sage: A2.pst(1,2) == Sq(4)*Sq(2) + Sq(2)*Sq(4) True sage: SteenrodAlgebra(5).pst(3,5) P(0,0,0,0,125) """ raise ValueError("%s is not a non-negative integer" % s) raise ValueError("%s is not a positive integer" % t)
def ngens(self): r""" Number of generators of self.
OUTPUT: number or Infinity
The Steenrod algebra is infinitely generated. A sub-Hopf algebra may be finitely or infinitely generated; in general, it is not clear what a minimal generating set is, nor the cardinality of that set. So: if the algebra is infinite-dimensional, this returns Infinity. If the algebra is finite-dimensional and is equal to one of the sub-Hopf algebras `A(n)`, then their minimal generating set is known, and this returns the cardinality of that set. Otherwise, any sub-Hopf algebra is (not necessarily minimally) generated by the `P^s_t`'s that it contains (along with the `Q_n`'s it contains, at odd primes), so this returns the number of `P^s_t`'s and `Q_n`'s in the algebra.
EXAMPLES::
sage: A = SteenrodAlgebra(3) sage: A.ngens() +Infinity sage: SteenrodAlgebra(profile=lambda n: n).ngens() +Infinity sage: SteenrodAlgebra(profile=[3,2,1]).ngens() # A(2) 3 sage: SteenrodAlgebra(profile=[3,2,1], basis='pst').ngens() 3 sage: SteenrodAlgebra(p=3, profile=[[3,2,1], [2,2,2,2]]).ngens() # A(3) at p=3 4 sage: SteenrodAlgebra(profile=[1,2,1,1]).ngens() 5 """ return sum(self._profile[0]) + len([a for a in self._profile[1] if a == 2])
def gens(self): r""" Family of generators for this algebra.
OUTPUT: family of elements of this algebra
At the prime 2, the Steenrod algebra is generated by the elements `\text{Sq}^{2^i}` for `i \geq 0`. At odd primes, it is generated by the elements `Q_0` and `\mathcal{P}^{p^i}` for `i \geq 0`. So if this algebra is the entire Steenrod algebra, return an infinite family made up of these elements.
For sub-Hopf algebras of the Steenrod algebra, it is not always clear what a minimal generating set is. The sub-Hopf algebra `A(n)` is minimally generated by the elements `\text{Sq}^{2^i}` for `0 \leq i \leq n` at the prime 2. At odd primes, `A(n)` is minimally generated by `Q_0` along with `\mathcal{P}^{p^i}` for `0 \leq i \leq n-1`. So if this algebra is `A(n)`, return the appropriate list of generators.
For other sub-Hopf algebras: return a non-minimal generating set: the family of `P^s_t`'s and `Q_n`'s contained in the algebra.
EXAMPLES::
sage: A3 = SteenrodAlgebra(3, 'adem') sage: A3.gens() Lazy family (<bound method SteenrodAlgebra_generic_with_category.gen of mod 3 Steenrod algebra, serre-cartan basis>(i))_{i in Non negative integers} sage: A3.gens()[0] beta sage: A3.gens()[1] P^1 sage: A3.gens()[2] P^3 sage: SteenrodAlgebra(profile=[3,2,1]).gens() Family (Sq(1), Sq(2), Sq(4))
In the following case, return a non-minimal generating set. (It is not minimal because `\text{Sq}(0,0,1)` is the commutator of `\text{Sq}(1)` and `\text{Sq}(0,2)`.) ::
sage: SteenrodAlgebra(profile=[1,2,1]).gens() Family (Sq(1), Sq(0,1), Sq(0,2), Sq(0,0,1)) sage: SteenrodAlgebra(p=5, profile=[[2,1], [2,2,2]]).gens() Family (Q_0, P(1), P(5)) sage: SteenrodAlgebra(profile=lambda n: n).gens() Lazy family (<bound method SteenrodAlgebra_mod_two_with_category.gen of sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [1, 2, 3, ..., 98, 99, +Infinity, +Infinity, +Infinity, ...]>(i))_{i in Non negative integers}
You may also use ``algebra_generators`` instead of ``gens``::
sage: SteenrodAlgebra(p=5, profile=[[2,1], [2,2,2]]).algebra_generators() Family (Q_0, P(1), P(5)) """
algebra_generators = gens
def gen(self, i=0): r""" The ith generator of this algebra.
INPUT:
- ``i`` - non-negative integer
OUTPUT: the ith generator of this algebra
For the full Steenrod algebra, the `i^{th}` generator is `\text{Sq}(2^i)` at the prime 2; when `p` is odd, the 0th generator is `\beta = Q(0)`, and for `i>0`, the `i^{th}` generator is `P(p^{i-1})`.
For sub-Hopf algebras of the Steenrod algebra, it is not always clear what a minimal generating set is. The sub-Hopf algebra `A(n)` is minimally generated by the elements `\text{Sq}^{2^i}` for `0 \leq i \leq n` at the prime 2. At odd primes, `A(n)` is minimally generated by `Q_0` along with `\mathcal{P}^{p^i}` for `0 \leq i \leq n-1`. So if this algebra is `A(n)`, return the appropriate generator.
For other sub-Hopf algebras: they are generated (but not necessarily minimally) by the `P^s_t`'s (and `Q_n`'s, if `p` is odd) that they contain. So order the `P^s_t`'s (and `Q_n`'s) in the algebra by degree and return the `i`-th one.
EXAMPLES::
sage: A = SteenrodAlgebra(2) sage: A.gen(4) Sq(16) sage: A.gen(200) Sq(1606938044258990275541962092341162602522202993782792835301376) sage: SteenrodAlgebra(2, basis='adem').gen(2) Sq^4 sage: SteenrodAlgebra(2, basis='pst').gen(2) P^2_1 sage: B = SteenrodAlgebra(5) sage: B.gen(0) Q_0 sage: B.gen(2) P(5)
sage: SteenrodAlgebra(profile=[2,1]).gen(1) Sq(2) sage: SteenrodAlgebra(profile=[1,2,1]).gen(1) Sq(0,1) sage: SteenrodAlgebra(profile=[1,2,1]).gen(5) Traceback (most recent call last): ... ValueError: This algebra only has 4 generators, so call gen(i) with 0 <= i < 4
sage: D = SteenrodAlgebra(profile=lambda n: n) sage: [D.gen(n) for n in range(5)] [Sq(1), Sq(0,1), Sq(0,2), Sq(0,0,1), Sq(0,0,2)] sage: D3 = SteenrodAlgebra(p=3, profile=(lambda n: n, lambda n: 2)) sage: [D3.gen(n) for n in range(9)] [Q_0, P(1), Q_1, P(0,1), Q_2, P(0,3), P(0,0,1), Q_3, P(0,0,3)] sage: D3 = SteenrodAlgebra(p=3, profile=(lambda n: n, lambda n: 1 if n<1 else 2)) sage: [D3.gen(n) for n in range(9)] [P(1), Q_1, P(0,1), Q_2, P(0,3), P(0,0,1), Q_3, P(0,0,3), P(0,0,0,1)] sage: SteenrodAlgebra(p=5, profile=[[2,1], [2,2,2]], basis='pst').gen(2) P^1_1 """ raise ValueError("%s is not a non-negative integer" % i) # check to see if equal to A(n) for some n. # if not A(n), return list of P^s_t's in algebra, along with Q's if p is odd else: last_t = max(len(self._profile[0]), len(self._profile[1])) guess = self.Q(j-1) idx += 1
# entire Steenrod algebra: else:
# infinite-dimensional sub-Hopf algebra
def is_commutative(self): r""" True if ``self`` is graded commutative, as determined by the profile function. In particular, a sub-Hopf algebra of the mod 2 Steenrod algebra is commutative if and only if there is an integer `n>0` so that its profile function `e` satisfies
- `e(i) = 0` for `i < n`, - `e(i) \leq n` for `i \geq n`.
When `p` is odd, there must be an integer `n \geq 0` so that the profile functions `e` and `k` satisfy
- `e(i) = 0` for `i < n`, - `e(i) \leq n` for `i \geq n`. - `k(i) = 1` for `i < n`.
EXAMPLES::
sage: A = SteenrodAlgebra(p=3) sage: A.is_commutative() False sage: SteenrodAlgebra(profile=[2,1]).is_commutative() False sage: SteenrodAlgebra(profile=[0,2,2,1]).is_commutative() True
Note that if the profile function is specified by a function, then by default it has infinite truncation type: the profile function is assumed to be infinite after the 100th term. ::
sage: SteenrodAlgebra(profile=lambda n: 1).is_commutative() False sage: SteenrodAlgebra(profile=lambda n: 1, truncation_type=0).is_commutative() True
sage: SteenrodAlgebra(p=5, profile=([0,2,2,1], [])).is_commutative() True sage: SteenrodAlgebra(p=5, profile=([0,2,2,1], [1,1,2])).is_commutative() True sage: SteenrodAlgebra(p=5, profile=([0,2,1], [1,2,2,2])).is_commutative() False """ and all([self.profile(i,1) == 1 for i in range(n)]))
def is_finite(self): r""" True if this algebra is finite-dimensional.
Therefore true if the profile function is finite, and in particular the ``truncation_type`` must be finite.
EXAMPLES::
sage: A = SteenrodAlgebra(p=3) sage: A.is_finite() False sage: SteenrodAlgebra(profile=[3,2,1]).is_finite() True sage: SteenrodAlgebra(profile=lambda n: n).is_finite() False """
def dimension(self): r""" The dimension of this algebra as a vector space over `\GF{p}`.
If the algebra is infinite, return ``+Infinity``. Otherwise, the profile function must be finite. In this case, at the prime 2, its dimension is `2^s`, where `s` is the sum of the entries in the profile function. At odd primes, the dimension is `p^s * 2^t` where `s` is the sum of the `e` component of the profile function and `t` is the number of 2's in the `k` component of the profile function.
EXAMPLES::
sage: SteenrodAlgebra(p=7).dimension() +Infinity sage: SteenrodAlgebra(profile=[3,2,1]).dimension() 64 sage: SteenrodAlgebra(p=3, profile=([1,1], [])).dimension() 9 sage: SteenrodAlgebra(p=5, profile=([1], [2,2])).dimension() 20 """
@cached_method def top_class(self): r""" Highest dimensional basis element. This is only defined if the algebra is finite.
EXAMPLES::
sage: SteenrodAlgebra(2,profile=(3,2,1)).top_class() Sq(7,3,1) sage: SteenrodAlgebra(3,profile=((2,2,1),(1,2,2,2,2))).top_class() Q_1 Q_2 Q_3 Q_4 P(8,8,2)
TESTS::
sage: SteenrodAlgebra(2,profile=(3,2,1),basis='pst').top_class() P^0_1 P^0_2 P^1_1 P^0_3 P^1_2 P^2_1 sage: SteenrodAlgebra(5,profile=((0,),(2,1,2,2))).top_class() Q_0 Q_2 Q_3 sage: SteenrodAlgebra(5).top_class() Traceback (most recent call last): ... ValueError: the algebra is not finite dimensional
Currently, we create the top class in the Milnor basis version and transform this result back into the requested basis. This approach is easy to implement but far from optimal for the 'pst' basis. Occasionally, it also gives an awkward leading coefficient::
sage: SteenrodAlgebra(3,profile=((2,1),(1,2,2)),basis='pst').top_class() 2 Q_1 Q_2 (P^0_1)^2 (P^0_2)^2 (P^1_1)^2
TESTS::
sage: A=SteenrodAlgebra(2,profile=(3,2,1),basis='pst') sage: A.top_class().parent() is A True """ # we create the top class in the Milnor basis version else:
def order(self): r""" The order of this algebra.
This is computed by computing its vector space dimension `d` and then returning `p^d`.
EXAMPLES::
sage: SteenrodAlgebra(p=7).order() +Infinity sage: SteenrodAlgebra(profile=[2,1]).dimension() 8 sage: SteenrodAlgebra(profile=[2,1]).order() 256 sage: SteenrodAlgebra(p=3, profile=([1], [])).dimension() 3 sage: SteenrodAlgebra(p=3, profile=([1], [])).order() 27 sage: SteenrodAlgebra(p=5, profile=([], [2, 2])).dimension() 4 sage: SteenrodAlgebra(p=5, profile=([], [2, 2])).order() == 5**4 True """
def is_division_algebra(self): r""" The only way this algebra can be a division algebra is if it is the ground field `\GF{p}`.
EXAMPLES::
sage: SteenrodAlgebra(11).is_division_algebra() False sage: SteenrodAlgebra(profile=lambda n: 0, truncation_type=0).is_division_algebra() True """
def is_field(self, proof = True): r""" The only way this algebra can be a field is if it is the ground field `\GF{p}`.
EXAMPLES::
sage: SteenrodAlgebra(11).is_field() False sage: SteenrodAlgebra(profile=lambda n: 0, truncation_type=0).is_field() True """
def is_integral_domain(self, proof = True): r""" The only way this algebra can be an integral domain is if it is the ground field `\GF{p}`.
EXAMPLES::
sage: SteenrodAlgebra(11).is_integral_domain() False sage: SteenrodAlgebra(profile=lambda n: 0, truncation_type=0).is_integral_domain() True """
def is_noetherian(self): """ This algebra is noetherian if and only if it is finite.
EXAMPLES::
sage: SteenrodAlgebra(3).is_noetherian() False sage: SteenrodAlgebra(profile=[1,2,1]).is_noetherian() True sage: SteenrodAlgebra(profile=lambda n: n+2).is_noetherian() False """
def is_generic(self): r""" The algebra is generic if it is based on the odd-primary relations, i.e. if its dual is a quotient of
.. MATH::
A_* = \GF{p} [\xi_1, \xi_2, \xi_3, ...] \otimes \Lambda (\tau_0, \tau_1, ...)
Sage also allows this for `p=2`. Only the usual Steenrod algebra at the prime `2` and its sub algebras are non-generic.
EXAMPLES::
sage: SteenrodAlgebra(3).is_generic() True sage: SteenrodAlgebra(2).is_generic() False sage: SteenrodAlgebra(2,generic=True).is_generic() True """
###################################################### # element class ######################################################
class Element(CombinatorialFreeModule.Element): r""" Class for elements of the Steenrod algebra. Since the Steenrod algebra class is based on :class:`CombinatorialFreeModule <sage.combinat.free_module.CombinatorialFreeModule>`, this is based on :class:`IndexedFreeModuleElement <sage.modules.with_basis.indexed_element.IndexedFreeModuleElement>`. It has new methods reflecting its role, like :meth:`degree` for computing the degree of an element.
EXAMPLES:
Since this class inherits from :class:`IndexedFreeModuleElement <sage.modules.with_basis.indexed_element.IndexedFreeModuleElement>`, elements can be used as iterators, and there are other useful methods::
sage: c = Sq(5).antipode(); c Sq(2,1) + Sq(5) sage: for mono, coeff in c: print((coeff, mono)) (1, (5,)) (1, (2, 1)) sage: c.monomial_coefficients() {(2, 1): 1, (5,): 1} sage: sorted(c.monomials(), key=lambda x: x.support()) [Sq(2,1), Sq(5)] sage: sorted(c.support()) [(2, 1), (5,)]
See the documentation for this module (type ``sage.algebras.steenrod.steenrod_algebra?``) for more information about elements of the Steenrod algebra. """ def prime(self): """ The prime associated to self.
EXAMPLES::
sage: a = SteenrodAlgebra().Sq(3,2,1) sage: a.prime() 2 sage: a.change_basis('adem').prime() 2 sage: b = SteenrodAlgebra(p=7).basis(36)[0] sage: b.prime() 7 sage: SteenrodAlgebra(p=3, basis='adem').one().prime() 3 """
def basis_name(self): """ The basis name associated to self.
EXAMPLES::
sage: a = SteenrodAlgebra().Sq(3,2,1) sage: a.basis_name() 'milnor' sage: a.change_basis('adem').basis_name() 'serre-cartan' sage: a.change_basis('wood____y').basis_name() 'woody' sage: b = SteenrodAlgebra(p=7).basis(36)[0] sage: b.basis_name() 'milnor' sage: a.change_basis('adem').basis_name() 'serre-cartan' """
def is_homogeneous(self): """ Return True iff this element is homogeneous.
EXAMPLES::
sage: (Sq(0,0,1) + Sq(7)).is_homogeneous() True sage: (Sq(0,0,1) + Sq(2)).is_homogeneous() False """
def degree(self): r""" The degree of self.
The degree of `\text{Sq}(i_1,i_2,i_3,...)` is
.. MATH::
i_1 + 3i_2 + 7i_3 + ... + (2^k - 1) i_k + ....
At an odd prime `p`, the degree of `Q_k` is `2p^k - 1` and the degree of `\mathcal{P}(i_1, i_2, ...)` is
.. MATH::
\sum_{k \geq 0} 2(p^k - 1) i_k.
ALGORITHM: If :meth:`is_homogeneous` returns True, call :meth:`SteenrodAlgebra_generic.degree_on_basis` on the leading summand.
EXAMPLES::
sage: Sq(0,0,1).degree() 7 sage: (Sq(0,0,1) + Sq(7)).degree() 7 sage: (Sq(0,0,1) + Sq(2)).degree() Traceback (most recent call last): ... ValueError: Element is not homogeneous.
sage: A11 = SteenrodAlgebra(p=11) sage: A11.P(1).degree() 20 sage: A11.P(1,1).degree() 260 sage: A11.Q(2).degree() 241
TESTS::
sage: all([x.degree() == 10 for x in SteenrodAlgebra(basis='woody').basis(10)]) True sage: all([x.degree() == 11 for x in SteenrodAlgebra(basis='woodz').basis(11)]) True sage: all([x.degree() == x.milnor().degree() for x in SteenrodAlgebra(basis='wall').basis(11)]) True sage: a = SteenrodAlgebra(basis='pst').basis(10)[0] sage: a.degree() == a.change_basis('arnonc').degree() True sage: b = SteenrodAlgebra(basis='comm').basis(12)[1] sage: b.degree() == b.change_basis('adem').change_basis('arnona').degree() True sage: all([x.degree() == 9 for x in SteenrodAlgebra(basis='comm').basis(9)]) True sage: all([x.degree() == 8 for x in SteenrodAlgebra(basis='adem').basis(8)]) True sage: all([x.degree() == 7 for x in SteenrodAlgebra(basis='milnor').basis(7)]) True sage: all([x.degree() == 24 for x in SteenrodAlgebra(p=3).basis(24)]) True sage: all([x.degree() == 40 for x in SteenrodAlgebra(p=5, basis='serre-cartan').basis(40)]) True """ raise ValueError("The zero element does not have a well-defined degree.")
def milnor(self): """ Return this element in the Milnor basis; that is, as an element of the appropriate Steenrod algebra.
This just calls the method :meth:`SteenrodAlgebra_generic.milnor`.
EXAMPLES::
sage: Adem = SteenrodAlgebra(basis='adem') sage: a = Adem.basis(4)[1]; a Sq^3 Sq^1 sage: a.milnor() Sq(1,1) """
def change_basis(self, basis='milnor'): r""" Representation of element with respect to basis.
INPUT:
- ``basis`` - string, basis in which to work.
OUTPUT: representation of self in given basis
The choices for ``basis`` are:
- 'milnor' for the Milnor basis. - 'serre-cartan', 'serre_cartan', 'sc', 'adem', 'admissible' for the Serre-Cartan basis. - 'wood_y' for Wood's Y basis. - 'wood_z' for Wood's Z basis. - 'wall' for Wall's basis. - 'wall_long' for Wall's basis, alternate representation - 'arnon_a' for Arnon's A basis. - 'arnon_a_long' for Arnon's A basis, alternate representation. - 'arnon_c' for Arnon's C basis. - 'pst', 'pst_rlex', 'pst_llex', 'pst_deg', 'pst_revz' for various `P^s_t`-bases. - 'comm', 'comm_rlex', 'comm_llex', 'comm_deg', 'comm_revz' for various commutator bases. - 'comm_long', 'comm_rlex_long', etc., for commutator bases, alternate representations.
See documentation for this module (by browsing the reference manual or by typing ``sage.algebras.steenrod.steenrod_algebra?``) for descriptions of the different bases.
EXAMPLES::
sage: c = Sq(2) * Sq(1) sage: c.change_basis('milnor') Sq(0,1) + Sq(3) sage: c.change_basis('serre-cartan') Sq^2 Sq^1 sage: d = Sq(0,0,1) sage: d.change_basis('arnonc') Sq^2 Sq^5 + Sq^4 Sq^2 Sq^1 + Sq^4 Sq^3 + Sq^7 """
def _basis_dictionary(self,basis): r""" Convert self to ``basis``, returning a dictionary of terms of the form (mono: coeff), where mono is a monomial in the given basis.
INPUT:
- ``basis`` - string, basis in which to work
OUTPUT: dictionary
This just calls :meth:`change_basis` to get an element of the Steenrod algebra with the new basis, and then calls :meth:`monomial_coefficients` on this element to produce its dictionary representation.
EXAMPLES::
sage: c = Sq(2) * Sq(1) sage: c._basis_dictionary('milnor') {(0, 1): 1, (3,): 1} sage: c Sq(0,1) + Sq(3) sage: c._basis_dictionary('serre-cartan') {(2, 1): 1} sage: c.change_basis('serre-cartan') Sq^2 Sq^1 sage: d = Sq(0,0,1) sage: d._basis_dictionary('arnonc') {(2, 5): 1, (4, 2, 1): 1, (4, 3): 1, (7,): 1} sage: d.change_basis('arnonc') Sq^2 Sq^5 + Sq^4 Sq^2 Sq^1 + Sq^4 Sq^3 + Sq^7
At odd primes::
sage: e = 2 * SteenrodAlgebra(3).P(1,2) sage: e._basis_dictionary('milnor') {((), (1, 2)): 2} sage: e 2 P(1,2) sage: e._basis_dictionary('serre-cartan') {(0, 7, 0, 2, 0): 2, (0, 8, 0, 1, 0): 2} sage: e.change_basis('adem') 2 P^7 P^2 + 2 P^8 P^1 """
def coproduct(self, algorithm='milnor'): """ The coproduct of this element.
INPUT:
- ``algorithm`` -- ``None`` or a string, either 'milnor' or 'serre-cartan' (or anything which will be converted to one of these by the function :func:`get_basis_name <sage.algebras.steenrod.steenrod_algebra_misc.get_basis_name>`). If ``None``, default to 'serre-cartan' if current basis is 'serre-cartan'; otherwise use 'milnor'.
See :meth:`SteenrodAlgebra_generic.coproduct_on_basis` for more information on computing the coproduct.
EXAMPLES::
sage: a = Sq(2) sage: a.coproduct() 1 # Sq(2) + Sq(1) # Sq(1) + Sq(2) # 1 sage: b = Sq(4) sage: (a*b).coproduct() == (a.coproduct()) * (b.coproduct()) True
sage: c = a.change_basis('adem'); c.coproduct(algorithm='milnor') 1 # Sq^2 + Sq^1 # Sq^1 + Sq^2 # 1 sage: c = a.change_basis('adem'); c.coproduct(algorithm='adem') 1 # Sq^2 + Sq^1 # Sq^1 + Sq^2 # 1
sage: d = a.change_basis('comm_long'); d.coproduct() 1 # s_2 + s_1 # s_1 + s_2 # 1
sage: A7 = SteenrodAlgebra(p=7) sage: a = A7.Q(1) * A7.P(1); a Q_1 P(1) sage: a.coproduct() 1 # Q_1 P(1) + P(1) # Q_1 + Q_1 # P(1) + Q_1 P(1) # 1 sage: a.coproduct(algorithm='adem') 1 # Q_1 P(1) + P(1) # Q_1 + Q_1 # P(1) + Q_1 P(1) # 1 """
def excess(self): r""" Excess of element.
OUTPUT: ``excess`` - non-negative integer
The excess of a Milnor basis element `\text{Sq}(a,b,c,...)` is `a + b + c + \cdots`. When `p` is odd, the excess of `Q_{0}^{e_0} Q_{1}^{e_1} \cdots P(r_1, r_2, ...)` is `\sum e_i + 2 \sum r_i`. The excess of a linear combination of Milnor basis elements is the minimum of the excesses of those basis elements.
See [Kr1971]_ for the proofs of these assertions.
EXAMPLES::
sage: a = Sq(1,2,3) sage: a.excess() 6 sage: (Sq(0,0,1) + Sq(4,1) + Sq(7)).excess() 1 sage: elt = Sq(0,0,1) + Sq(4,1) + Sq(7) sage: M = sorted(elt.monomials(), key=lambda x: x.support()) sage: [m.excess() for m in M] [1, 5, 7] sage: [m for m in M] [Sq(0,0,1), Sq(4,1), Sq(7)] sage: B = SteenrodAlgebra(7) sage: a = B.Q(1,2,5) sage: b = B.P(2,2,3) sage: a.excess() 3 sage: b.excess() 14 sage: (a + b).excess() 3 sage: (a * b).excess() 17 """ """ Excess of mono, where mono has the form ((s0, s1, ...), (r1, r2, ...)).
Returns the length of the first component, since that is the number of factors, plus twice the sum of the terms in the second component. """ return 0 else:
else:
def is_unit(self): """ True if element has a nonzero scalar multiple of P(0) as a summand, False otherwise.
EXAMPLES::
sage: z = Sq(4,2) + Sq(7,1) + Sq(3,0,1) sage: z.is_unit() False sage: u = Sq(0) + Sq(3,1) sage: u == 1 + Sq(3,1) True sage: u.is_unit() True sage: A5 = SteenrodAlgebra(5) sage: v = A5.P(0) sage: (v + v + v).is_unit() True """
def is_nilpotent(self): """ True if element is not a unit, False otherwise.
EXAMPLES::
sage: z = Sq(4,2) + Sq(7,1) + Sq(3,0,1) sage: z.is_nilpotent() True sage: u = 1 + Sq(3,1) sage: u == 1 + Sq(3,1) True sage: u.is_nilpotent() False """
def may_weight(self): r""" May's 'weight' of element.
OUTPUT: ``weight`` - non-negative integer
If we let `F_* (A)` be the May filtration of the Steenrod algebra, the weight of an element `x` is the integer `k` so that `x` is in `F_k(A)` and not in `F_{k+1}(A)`. According to Theorem 2.6 in May's thesis [May1964]_, the weight of a Milnor basis element is computed as follows: first, to compute the weight of `P(r_1,r_2, ...)`, write each `r_i` in base `p` as `r_i = \sum_j p^j r_{ij}`. Then each nonzero binary digit `r_{ij}` contributes `i` to the weight: the weight is `\sum_{i,j} i r_{ij}`. When `p` is odd, the weight of `Q_i` is `i+1`, so the weight of a product `Q_{i_1} Q_{i_2} ...` equals `(i_1+1) + (i_2+1) + ...`. Then the weight of `Q_{i_1} Q_{i_2} ...P(r_1,r_2, ...)` is the sum of `(i_1+1) + (i_2+1) + ...` and `\sum_{i,j} i r_{ij}`.
The weight of a sum of Milnor basis elements is the minimum of the weights of the summands.
When `p=2`, we compute the weight on Milnor basis elements by adding up the terms in their 'height' - see :meth:`wall_height` for documentation. (When `p` is odd, the height of an element is not defined.)
EXAMPLES::
sage: Sq(0).may_weight() 0 sage: a = Sq(4) sage: a.may_weight() 1 sage: b = Sq(4)*Sq(8) + Sq(8)*Sq(4) sage: b.may_weight() 2 sage: Sq(2,1,5).wall_height() [2, 3, 2, 1, 1] sage: Sq(2,1,5).may_weight() 9 sage: A5 = SteenrodAlgebra(5) sage: a = A5.Q(1,2,4) sage: b = A5.P(1,2,1) sage: a.may_weight() 10 sage: b.may_weight() 8 sage: (a * b).may_weight() 18 sage: A5.P(0,0,1).may_weight() 3 """ return Infinity else: # p odd
def is_decomposable(self): r""" Return True if element is decomposable, False otherwise. That is, if element is in the square of the augmentation ideal, return True; otherwise, return False.
OUTPUT: boolean
EXAMPLES::
sage: a = Sq(6) sage: a.is_decomposable() True sage: for i in range(9): ....: if not Sq(i).is_decomposable(): ....: print(Sq(i)) 1 Sq(1) Sq(2) Sq(4) Sq(8) sage: A3 = SteenrodAlgebra(p=3, basis='adem') sage: [A3.P(n) for n in range(30) if not A3.P(n).is_decomposable()] [1, P^1, P^3, P^9, P^27]
TESTS:
These all test changing bases and printing in various bases::
sage: A = SteenrodAlgebra(basis='milnor') sage: [A.Sq(n) for n in range(9) if not A.Sq(n).is_decomposable()] [1, Sq(1), Sq(2), Sq(4), Sq(8)] sage: A = SteenrodAlgebra(basis='wall_long') sage: [A.Sq(n) for n in range(9) if not A.Sq(n).is_decomposable()] [1, Sq^1, Sq^2, Sq^4, Sq^8] sage: A = SteenrodAlgebra(basis='arnona_long') sage: [A.Sq(n) for n in range(9) if not A.Sq(n).is_decomposable()] [1, Sq^1, Sq^2, Sq^4, Sq^8] sage: A = SteenrodAlgebra(basis='woodz') sage: [A.Sq(n) for n in range(20) if not A.Sq(n).is_decomposable()] # long time [1, Sq^1, Sq^2, Sq^4, Sq^8, Sq^16] sage: A = SteenrodAlgebra(basis='comm_long') sage: [A.Sq(n) for n in range(25) if not A.Sq(n).is_decomposable()] # long time [1, s_1, s_2, s_4, s_8, s_16] """
def wall_height(self): r""" Wall's 'height' of element.
OUTPUT: list of non-negative integers
The height of an element of the mod 2 Steenrod algebra is a list of non-negative integers, defined as follows: if the element is a monomial in the generators `\text{Sq}(2^i)`, then the `i^{th}` entry in the list is the number of times `\text{Sq}(2^i)` appears. For an arbitrary element, write it as a sum of such monomials; then its height is the maximum, ordered right-lexicographically, of the heights of those monomials.
When `p` is odd, the height of an element is not defined.
According to Theorem 3 in [Wal1960]_, the height of the Milnor basis element `\text{Sq}(r_1, r_2, ...)` is obtained as follows: write each `r_i` in binary as `r_i = \sum_j 2^j r_{ij}`. Then each nonzero binary digit `r_{ij}` contributes 1 to the `k^{th}` entry in the height, for `j \leq k \leq i+j-1`.
EXAMPLES::
sage: Sq(0).wall_height() [] sage: a = Sq(4) sage: a.wall_height() [0, 0, 1] sage: b = Sq(4)*Sq(8) + Sq(8)*Sq(4) sage: b.wall_height() [0, 0, 1, 1] sage: Sq(0,0,3).wall_height() [1, 2, 2, 1] """ raise NotImplementedError("Wall height is not defined at odd primes.")
def additive_order(self): """ The additive order of any nonzero element of the mod p Steenrod algebra is p.
OUTPUT: 1 (for the zero element) or p (for anything else)
EXAMPLES::
sage: z = Sq(4) + Sq(6) + 1 sage: z.additive_order() 2 sage: (Sq(3) + Sq(3)).additive_order() 1 """
class SteenrodAlgebra_mod_two(SteenrodAlgebra_generic): """ The mod 2 Steenrod algebra.
Users should not call this, but use the function :func:`SteenrodAlgebra` instead. See that function for extensive documentation. (This differs from :class:`SteenrodAlgebra_generic` only in that it has a method :meth:`Sq` for defining elements.) """ def Sq(self, *nums): r""" Milnor element `\text{Sq}(a,b,c,...)`.
INPUT:
- ``a, b, c, ...`` - non-negative integers
OUTPUT: element of the Steenrod algebra
This returns the Milnor basis element `\text{Sq}(a, b, c, ...)`.
EXAMPLES::
sage: A = SteenrodAlgebra(2) sage: A.Sq(5) Sq(5) sage: A.Sq(5,0,2) Sq(5,0,2)
Entries must be non-negative integers; otherwise, an error results. """ else: raise ValueError("Sq is only defined at the prime 2")
def SteenrodAlgebra(p=2, basis='milnor', generic='auto', **kwds): r""" The mod `p` Steenrod algebra
INPUT:
- ``p`` - positive prime integer (optional, default = 2) - ``basis`` - string (optional, default = 'milnor') - ``profile`` - a profile function in form specified below (optional, default ``None``) - ``truncation_type`` - 0 or `\infty` or 'auto' (optional, default 'auto') - ``precision`` - integer or ``None`` (optional, default ``None``) - ``generic`` - (optional, default 'auto')
OUTPUT: mod `p` Steenrod algebra or one of its sub-Hopf algebras, elements of which are printed using ``basis``
See below for information about ``basis``, ``profile``, etc.
EXAMPLES:
Some properties of the Steenrod algebra are available::
sage: A = SteenrodAlgebra(2) sage: A.order() +Infinity sage: A.is_finite() False sage: A.is_commutative() False sage: A.is_noetherian() False sage: A.is_integral_domain() False sage: A.is_field() False sage: A.is_division_algebra() False sage: A.category() Category of graded hopf algebras with basis over Finite Field of size 2
There are methods for constructing elements of the Steenrod algebra::
sage: A2 = SteenrodAlgebra(2); A2 mod 2 Steenrod algebra, milnor basis sage: A2.Sq(1,2,6) Sq(1,2,6) sage: A2.Q(3,4) # product of Milnor primitives Q_3 and Q_4 Sq(0,0,0,1,1) sage: A2.pst(2,3) # Margolis pst element Sq(0,0,4) sage: A5 = SteenrodAlgebra(5); A5 mod 5 Steenrod algebra, milnor basis sage: A5.P(1,2,6) P(1,2,6) sage: A5.Q(3,4) Q_3 Q_4 sage: A5.Q(3,4) * A5.P(1,2,6) Q_3 Q_4 P(1,2,6) sage: A5.pst(2,3) P(0,0,25)
You can test whether elements are contained in the Steenrod algebra::
sage: w = Sq(2) * Sq(4) sage: w in SteenrodAlgebra(2) True sage: w in SteenrodAlgebra(17) False
.. rubric:: Different bases for the Steenrod algebra:
There are two standard vector space bases for the mod `p` Steenrod algebra: the Milnor basis and the Serre-Cartan basis. When `p=2`, there are also several other, less well-known, bases. See the documentation for this module (type ``sage.algebras.steenrod.steenrod_algebra?``) and the function :func:`steenrod_algebra_basis <sage.algebras.steenrod.steenrod_algebra_bases.steenrod_algebra_basis_>` for full descriptions of each of the implemented bases.
This module implements the following bases at all primes:
- 'milnor': Milnor basis.
- 'serre-cartan' or 'adem' or 'admissible': Serre-Cartan basis.
- 'pst', 'pst_rlex', 'pst_llex', 'pst_deg', 'pst_revz': various `P^s_t`-bases.
- 'comm', 'comm_rlex', 'comm_llex', 'comm_deg', 'comm_revz', or these with '_long' appended: various commutator bases.
It implements the following bases when `p=2`:
- 'wood_y': Wood's Y basis.
- 'wood_z': Wood's Z basis.
- 'wall', 'wall_long': Wall's basis.
- 'arnon_a', 'arnon_a_long': Arnon's A basis.
- 'arnon_c': Arnon's C basis.
When defining a Steenrod algebra, you can specify a basis. Then elements of that Steenrod algebra are printed in that basis::
sage: adem = SteenrodAlgebra(2, 'adem') sage: x = adem.Sq(2,1) # Sq(-) always means a Milnor basis element sage: x Sq^4 Sq^1 + Sq^5 sage: y = Sq(0,1) # unadorned Sq defines elements w.r.t. Milnor basis sage: y Sq(0,1) sage: adem(y) Sq^2 Sq^1 + Sq^3 sage: adem5 = SteenrodAlgebra(5, 'serre-cartan') sage: adem5.P(0,2) P^10 P^2 + 4 P^11 P^1 + P^12
If you add or multiply elements defined using different bases, the left-hand factor determines the form of the output::
sage: SteenrodAlgebra(basis='adem').Sq(3) + SteenrodAlgebra(basis='pst').Sq(0,1) Sq^2 Sq^1 sage: SteenrodAlgebra(basis='pst').Sq(3) + SteenrodAlgebra(basis='milnor').Sq(0,1) P^0_1 P^1_1 + P^0_2 sage: SteenrodAlgebra(basis='milnor').Sq(2) * SteenrodAlgebra(basis='arnonc').Sq(2) Sq(1,1)
You can get a list of basis elements in a given dimension::
sage: A3 = SteenrodAlgebra(3, 'milnor') sage: A3.basis(13) Family (Q_1 P(2), Q_0 P(3))
Algebras defined over different bases are not equal::
sage: SteenrodAlgebra(basis='milnor') == SteenrodAlgebra(basis='pst') False
Bases have various synonyms, and in general Sage tries to figure out what basis you meant::
sage: SteenrodAlgebra(basis='MiLNOr') mod 2 Steenrod algebra, milnor basis sage: SteenrodAlgebra(basis='MiLNOr') == SteenrodAlgebra(basis='milnor') True sage: SteenrodAlgebra(basis='adem') mod 2 Steenrod algebra, serre-cartan basis sage: SteenrodAlgebra(basis='adem').basis_name() 'serre-cartan' sage: SteenrodAlgebra(basis='wood---z---').basis_name() 'woodz'
As noted above, several of the bases ('arnon_a', 'wall', 'comm') have alternate, sometimes longer, representations. These provide ways of expressing elements of the Steenrod algebra in terms of the `\text{Sq}^{2^n}`.
::
sage: A_long = SteenrodAlgebra(2, 'arnon_a_long') sage: A_long(Sq(6)) Sq^1 Sq^2 Sq^1 Sq^2 + Sq^2 Sq^4 sage: SteenrodAlgebra(2, 'wall_long')(Sq(6)) Sq^2 Sq^1 Sq^2 Sq^1 + Sq^2 Sq^4 sage: SteenrodAlgebra(2, 'comm_deg_long')(Sq(6)) s_1 s_2 s_12 + s_2 s_4
.. rubric:: Sub-Hopf algebras of the Steenrod algebra:
These are specified using the argument ``profile``, along with, optionally, ``truncation_type`` and ``precision``. The ``profile`` argument specifies the profile function for this algebra. Any sub-Hopf algebra of the Steenrod algebra is determined by its *profile function*. When `p=2`, this is a map `e` from the positive integers to the set of non-negative integers, plus `\infty`, corresponding to the sub-Hopf algebra dual to this quotient of the dual Steenrod algebra:
.. MATH::
\GF{2} [\xi_1, \xi_2, \xi_3, ...] / (\xi_1^{2^{e(1)}}, \xi_2^{2^{e(2)}}, \xi_3^{2^{e(3)}}, ...).
The profile function `e` must satisfy the condition
- `e(r) \geq \min( e(r-i) - i, e(i))` for all `0 < i < r`.
This is specified via ``profile``, and optionally ``precision`` and ``truncation_type``. First, ``profile`` must have one of the following forms:
- a list or tuple, e.g., ``[3,2,1]``, corresponding to the function sending 1 to 3, 2 to 2, 3 to 1, and all other integers to the value of ``truncation_type``. - a function from positive integers to non-negative integers (and `\infty`), e.g., ``lambda n: n+2``. - ``None`` or ``Infinity`` - use this for the profile function for the whole Steenrod algebra.
In the first and third cases, ``precision`` is ignored. In the second case, this function is converted to a tuple of length one less than ``precision``, which has default value 100. The function is truncated at this point, and all remaining values are set to the value of ``truncation_type``.
``truncation_type`` may be 0, `\infty`, or 'auto'. If it's 'auto', then it gets converted to 0 in the first case above (when ``profile`` is a list), and otherwise (when ``profile`` is a function, ``None``, or ``Infinity``) it gets converted to `\infty`.
For example, the sub-Hopf algebra `A(2)` has profile function ``[3,2,1,0,0,0,...]``, so it can be defined by any of the following::
sage: A2 = SteenrodAlgebra(profile=[3,2,1]) sage: B2 = SteenrodAlgebra(profile=[3,2,1,0,0]) # trailing 0's ignored sage: A2 == B2 True sage: C2 = SteenrodAlgebra(profile=lambda n: max(4-n, 0), truncation_type=0) sage: A2 == C2 True
In the following case, the profile function is specified by a function and ``truncation_type`` isn't specified, so it defaults to `\infty`; therefore this gives a different sub-Hopf algebra::
sage: D2 = SteenrodAlgebra(profile=lambda n: max(4-n, 0)) sage: A2 == D2 False sage: D2.is_finite() False sage: E2 = SteenrodAlgebra(profile=lambda n: max(4-n, 0), truncation_type=Infinity) sage: D2 == E2 True
The argument ``precision`` only needs to be specified if the profile function is defined by a function and you want to control when the profile switches from the given function to the truncation type. For example::
sage: D3 = SteenrodAlgebra(profile=lambda n: n, precision=3) sage: D3 sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [1, 2, +Infinity, +Infinity, +Infinity, ...] sage: D4 = SteenrodAlgebra(profile=lambda n: n, precision=4); D4 sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [1, 2, 3, +Infinity, +Infinity, +Infinity, ...] sage: D3 == D4 False
When `p` is odd, ``profile`` is a pair of functions `e` and `k`, corresponding to the quotient
.. MATH::
\GF{p} [\xi_1, \xi_2, \xi_3, ...] \otimes \Lambda (\tau_0, \tau_1, ...) / (\xi_1^{p^{e_1}}, \xi_2^{p^{e_2}}, ...; \tau_0^{k_0}, \tau_1^{k_1}, ...).
Together, the functions `e` and `k` must satisfy the conditions
- `e(r) \geq \min( e(r-i) - i, e(i))` for all `0 < i < r`,
- if `k(i+j) = 1`, then either `e(i) \leq j` or `k(j) = 1` for all `i \geq 1`, `j \geq 0`.
Therefore ``profile`` must have one of the following forms:
- a pair of lists or tuples, the second of which takes values in the set `\{1,2\}`, e.g., ``([3,2,1,1], [1,1,2,2,1])``.
- a pair of functions, one from the positive integers to non-negative integers (and `\infty`), one from the non-negative integers to the set `\{1,2\}`, e.g., ``(lambda n: n+2, lambda n: 1 if n<3 else 2)``.
- ``None`` or ``Infinity`` - use this for the profile function for the whole Steenrod algebra.
You can also mix and match the first two, passing a pair with first entry a list and second entry a function, for instance. The values of ``precision`` and ``truncation_type`` are determined by the first entry.
More examples::
sage: E = SteenrodAlgebra(profile=lambda n: 0 if n<3 else 3, truncation_type=0) sage: E.is_commutative() True
sage: A2 = SteenrodAlgebra(profile=[3,2,1]) # the algebra A(2) sage: Sq(7,3,1) in A2 True sage: Sq(8) in A2 False sage: Sq(8) in SteenrodAlgebra().basis(8) True sage: Sq(8) in A2.basis(8) False sage: A2.basis(8) Family (Sq(1,0,1), Sq(2,2), Sq(5,1))
sage: A5 = SteenrodAlgebra(p=5) sage: A51 = SteenrodAlgebra(p=5, profile=([1], [2,2])) sage: A5.Q(0,1) * A5.P(4) in A51 True sage: A5.Q(2) in A51 False sage: A5.P(5) in A51 False
For sub-Hopf algebras of the Steenrod algebra, only the Milnor basis or the various `P^s_t`-bases may be used. ::
sage: SteenrodAlgebra(profile=[1,2,1,1], basis='adem') Traceback (most recent call last): ... NotImplementedError: For sub-Hopf algebras of the Steenrod algebra, only the Milnor basis and the pst bases are implemented.
.. rubric:: The generic Steenrod algebra at the prime `2`:
The structure formulas for the Steenrod algebra at odd primes `p` also make sense when `p` is set to `2`. We refer to the resulting algebra as the "generic Steenrod algebra" for the prime `2`. The dual Hopf algebra is given by
.. MATH::
A_* = \GF{2} [\xi_1, \xi_2, \xi_3, ...] \otimes \Lambda (\tau_0, \tau_1, ...)
The degree of `\xi_k` is `2^{k+1}-2` and the degree of `\tau_k` is `2^{k+1}-1`.
The generic Steenrod algebra is an associated graded algebra of the usual Steenrod algebra that is occasionally useful. Its cohomology, for example, is the `E_2`-term of a spectral sequence that computes the `E_2`-term of the Novikov spectral sequence. It can also be obtained as a specialisation of Voevodsky's "motivic Steenrod algebra": in the notation of [Voe2003]_, Remark 12.12, it corresponds to setting `\rho = \tau = 0`. The usual Steenrod algebra is given by `\rho = 0` and `\tau = 1`.
In Sage this algebra is constructed using the 'generic' keyword.
Example::
sage: EA = SteenrodAlgebra(p=2,generic=True) ; EA generic mod 2 Steenrod algebra, milnor basis sage: EA[8] Vector space spanned by (Q_0 Q_2, Q_0 Q_1 P(2), P(1,1), P(4)) over Finite Field of size 2
TESTS:
Testing unique parents::
sage: S0 = SteenrodAlgebra(2) sage: S1 = SteenrodAlgebra(2) sage: S0 is S1 True sage: S2 = SteenrodAlgebra(2, basis='adem') sage: S0 is S2 False sage: S0 == S2 False sage: A1 = SteenrodAlgebra(profile=[2,1]) sage: B1 = SteenrodAlgebra(profile=[2,1,0,0]) sage: A1 is B1 True """ else:
def AA(n=None, p=2): r""" This returns the Steenrod algebra `A` or its sub-Hopf algebra `A(n)`.
INPUT:
- `n` - non-negative integer, optional (default ``None``) - `p` - prime number, optional (default 2)
OUTPUT: If `n` is ``None``, then return the full Steenrod algebra. Otherwise, return `A(n)`.
When `p=2`, `A(n)` is the sub-Hopf algebra generated by the elements `\text{Sq}^i` for `i \leq 2^n`. Its profile function is `(n+1, n, n-1, ...)`. When `p` is odd, `A(n)` is the sub-Hopf algebra generated by the elements `Q_0` and `\mathcal{P}^i` for `i \leq p^{n-1}`. Its profile function is `e=(n, n-1, n-2, ...)` and `k=(2, 2, ..., 2)` (length `n+1`).
EXAMPLES::
sage: from sage.algebras.steenrod.steenrod_algebra import AA as A sage: A() mod 2 Steenrod algebra, milnor basis sage: A(2) sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] sage: A(2, p=5) sub-Hopf algebra of mod 5 Steenrod algebra, milnor basis, profile function ([2, 1], [2, 2, 2]) """
def Sq(*nums): r""" Milnor element Sq(a,b,c,...).
INPUT:
- ``a, b, c, ...`` - non-negative integers
OUTPUT: element of the Steenrod algebra
This returns the Milnor basis element `\text{Sq}(a, b, c, ...)`.
EXAMPLES::
sage: Sq(5) Sq(5) sage: Sq(5) + Sq(2,1) + Sq(5) # addition is mod 2: Sq(2,1) sage: (Sq(4,3) + Sq(7,2)).degree() 13
Entries must be non-negative integers; otherwise, an error results.
This function is a good way to define elements of the Steenrod algebra. """ |