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r""" Examples of a Lie algebra """ #***************************************************************************** # Copyright (C) 2014 Travis Scrimshaw <tscrim at ucdavis.edu> # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #*****************************************************************************
#from sage.misc.cachefunc import cached_method
r""" An example of a Lie algebra: a Lie algebra generated by a set of elements of an associative algebra.
This class illustrates a minimal implementation of a Lie algebra.
Let `R` be a commutative ring, and `A` an associative `R`-algebra. The Lie algebra `A` (sometimes denoted `A^-`) is defined to be the `R`-module `A` with Lie bracket given by the commutator in `A`: that is, `[a, b] := ab - ba` for all `a, b \in A`.
What this class implements is not precisely `A^-`, however; it is the Lie subalgebra of `A^-` generated by the elements of the iterable ``gens``. This specific implementation does not provide a reasonable containment test (i.e., it does not allow you to check if a given element `a` of `A^-` belongs to this Lie subalgebra); it, however, allows computing inside it.
INPUT:
- ``gens`` -- a nonempty iterable consisting of elements of an associative algebra `A`
OUTPUT:
The Lie subalgebra of `A^-` generated by the elements of ``gens``
EXAMPLES:
We create a model of `\mathfrak{sl}_2` using matrices::
sage: gens = [matrix([[0,1],[0,0]]), matrix([[0,0],[1,0]]), matrix([[1,0],[0,-1]])] sage: for g in gens: ....: g.set_immutable() sage: L = LieAlgebras(QQ).example(gens) sage: e,f,h = L.lie_algebra_generators() sage: e.bracket(f) == h True sage: h.bracket(e) == 2*e True sage: h.bracket(f) == -2*f True """ def __classcall_private__(cls, gens): """ Normalize input to ensure a unique representation.
EXAMPLES::
sage: S3 = SymmetricGroupAlgebra(QQ, 3) sage: L1 = LieAlgebras(QQ).example() sage: gens = list(S3.algebra_generators()) sage: L2 = LieAlgebras(QQ).example(gens) sage: L1 is L2 True """
""" EXAMPLES::
sage: L = LieAlgebras(QQ).example() sage: TestSuite(L).run() """ raise ValueError("need at least one generator")
""" EXAMPLES::
sage: LieAlgebras(QQ).example() An example of a Lie algebra: the Lie algebra from the associative algebra Symmetric group algebra of order 3 over Rational Field generated by ([2, 1, 3], [2, 3, 1]) """ " associative algebra {} generated by {}".format( self._A, self._gens)
""" Return an element of ``self``.
EXAMPLES::
sage: S3 = SymmetricGroupAlgebra(ZZ, 3) sage: gens = S3.algebra_generators() sage: L = LieAlgebras(QQ).example() sage: L(3*gens[0] + gens[1]) 3*[2, 1, 3] + [2, 3, 1] """
""" Return the element 0.
EXAMPLES::
sage: L = LieAlgebras(QQ).example() sage: L.zero() 0 """
""" Return the generators of ``self`` as a Lie algebra.
EXAMPLES::
sage: L = LieAlgebras(QQ).example() sage: L.lie_algebra_generators() Family ([2, 1, 3], [2, 3, 1]) """
# TODO: refactor to use LieAlgebraElementWrapper once more of #14901 is added in """ Wrap an element as a Lie algebra element. """ """ Check equality.
This check is rather restrictive: ``self`` and ``rhs`` are only revealed as equal if they are equal *and* have the same parent (or both are zero).
EXAMPLES::
sage: L = LieAlgebras(QQ).example() sage: x,y = L.lie_algebra_generators() sage: x == x True sage: x.bracket(y) == -y.bracket(x) True sage: x == y False sage: x.bracket(x) == L.zero() True sage: x.bracket(x) == 0 True """
""" Check not-equals.
EXAMPLES::
sage: L = LieAlgebras(QQ).example() sage: x,y = L.lie_algebra_generators() sage: x != y True sage: x != 0 True sage: x != x False sage: x.bracket(y) != -y.bracket(x) False """
""" Check non-zero.
EXAMPLES::
sage: L = LieAlgebras(QQ).example() sage: bool(sum(L.lie_algebra_generators())) True sage: bool(L.zero()) False """
""" Add ``self`` and ``rhs``.
EXAMPLES::
sage: L = LieAlgebras(QQ).example() sage: x,y = L.lie_algebra_generators() sage: x + y [2, 1, 3] + [2, 3, 1] """
""" Subtract ``self`` and ``rhs``.
EXAMPLES::
sage: L = LieAlgebras(QQ).example() sage: x,y = L.lie_algebra_generators() sage: x - y [2, 1, 3] - [2, 3, 1] """
""" Return the action of a scalar on ``self``.
EXAMPLES::
sage: L = LieAlgebras(QQ).example() sage: x,y = L.lie_algebra_generators() sage: 3 * x 3*[2, 1, 3] """ # This was copied, but IDK if it still applies: # With the current design, the coercion model does not have # enough information to detect apriori that this method only # accepts scalars; so it tries on some elements(), and we need # to make sure to report an error. # Temporary needed by coercion (see Polynomial/FractionField tests). else: return None return self.__class__(self.parent(), self.value * scalar)
""" Division by coefficients.
EXAMPLES::
sage: L = LieAlgebras(QQ).example() sage: x,y = L.lie_algebra_generators() sage: y / 4 1/4*[2, 3, 1] """ return self * (~x)
""" Return the negation of ``self``.
EXAMPLES::
sage: L = LieAlgebras(QQ).example() sage: x,y = L.lie_algebra_generators() sage: -x -[2, 1, 3] """
""" Redirect the ``__getitem__()`` to the wrapped element.
EXAMPLES::
sage: gens = [matrix([[0,1],[0,0]]), matrix([[0,0],[1,0]]), matrix([[1,0],[0,-1]])] sage: for g in gens: ....: g.set_immutable() sage: L = LieAlgebras(QQ).example(gens) sage: e,f,h = L.lie_algebra_generators() sage: h[0,0] 1 sage: h[1,1] -1 sage: h[0,1] 0 """
""" Return the Lie bracket ``[self, rhs]``.
EXAMPLES::
sage: L = LieAlgebras(QQ).example() sage: x,y = L.lie_algebra_generators() sage: elt = 2*x - y sage: elt.bracket(elt) 0 sage: elt.bracket(x) -[1, 3, 2] + [3, 2, 1] sage: elt2 = x.bracket(y) + x sage: elt.bracket(elt2) -2*[2, 1, 3] + 4*[2, 3, 1] - 4*[3, 1, 2] + 2*[3, 2, 1] """
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