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# -*- coding: utf-8 -*- 

""" 

Free algebras 

 

AUTHORS: 

 

- David Kohel (2005-09) 

 

- William Stein (2006-11-01): add all doctests; implemented many 

things. 

 

- Simon King (2011-04): Put free algebras into the category framework. 

Reimplement free algebra constructor, using a 

:class:`~sage.structure.factory.UniqueFactory` for handling 

different implementations of free algebras. Allow degree weights 

for free algebras in letterplace implementation. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(ZZ,3,'x,y,z') 

sage: F.base_ring() 

Integer Ring 

sage: G = FreeAlgebra(F, 2, 'm,n'); G 

Free Algebra on 2 generators (m, n) over Free Algebra on 3 generators (x, y, z) over Integer Ring 

sage: G.base_ring() 

Free Algebra on 3 generators (x, y, z) over Integer Ring 

 

The above free algebra is based on a generic implementation. By 

:trac:`7797`, there is a different implementation 

:class:`~sage.algebras.letterplace.free_algebra_letterplace.FreeAlgebra_letterplace` 

based on Singular's letterplace rings. It is currently restricted to 

weighted homogeneous elements and is therefore not the default. But the 

arithmetic is much faster than in the generic implementation. 

Moreover, we can compute Groebner bases with degree bound for its 

two-sided ideals, and thus provide ideal containment tests:: 

 

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace') 

sage: F 

Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field 

sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F 

sage: I.groebner_basis(degbound=4) 

Twosided Ideal (y*z*y*y - y*z*y*z + y*z*z*y - y*z*z*z, y*z*y*x + y*z*y*z + y*z*z*x + y*z*z*z, y*y*z*y - y*y*z*z + y*z*z*y - y*z*z*z, y*y*z*x + y*y*z*z + y*z*z*x + y*z*z*z, y*y*y - y*y*z + y*z*y - y*z*z, y*y*x + y*y*z + y*z*x + y*z*z, x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field 

sage: y*z*y*y*z*z + 2*y*z*y*z*z*x + y*z*y*z*z*z - y*z*z*y*z*x + y*z*z*z*z*x in I 

True 

 

Positive integral degree weights for the letterplace implementation 

was introduced in :trac:`7797`:: 

 

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3]) 

sage: x.degree() 

2 

sage: y.degree() 

1 

sage: z.degree() 

3 

sage: I = F*[x*y-y*x, x^2+2*y*z, (x*y)^2-z^2]*F 

sage: Q.<a,b,c> = F.quo(I) 

sage: TestSuite(Q).run() 

sage: a^2*b^2 

c*c 

 

TESTS:: 

 

sage: F = FreeAlgebra(GF(5),3,'x') 

sage: TestSuite(F).run() 

sage: F is loads(dumps(F)) 

True 

sage: F = FreeAlgebra(GF(5),3,'x', implementation='letterplace') 

sage: TestSuite(F).run() 

sage: F is loads(dumps(F)) 

True 

 

:: 

 

sage: F.<x,y,z> = FreeAlgebra(GF(5),3) 

sage: TestSuite(F).run() 

sage: F is loads(dumps(F)) 

True 

sage: F.<x,y,z> = FreeAlgebra(GF(5),3, implementation='letterplace') 

sage: TestSuite(F).run() 

sage: F is loads(dumps(F)) 

True 

 

:: 

 

sage: F = FreeAlgebra(GF(5),3, ['xx', 'zba', 'Y']) 

sage: TestSuite(F).run() 

sage: F is loads(dumps(F)) 

True 

sage: F = FreeAlgebra(GF(5),3, ['xx', 'zba', 'Y'], implementation='letterplace') 

sage: TestSuite(F).run() 

sage: F is loads(dumps(F)) 

True 

 

:: 

 

sage: F = FreeAlgebra(GF(5),3, 'abc') 

sage: TestSuite(F).run() 

sage: F is loads(dumps(F)) 

True 

sage: F = FreeAlgebra(GF(5),3, 'abc', implementation='letterplace') 

sage: TestSuite(F).run() 

sage: F is loads(dumps(F)) 

True 

 

:: 

 

sage: F = FreeAlgebra(FreeAlgebra(ZZ,2,'ab'), 2, 'x') 

sage: TestSuite(F).run() 

sage: F is loads(dumps(F)) 

True 

 

Note that the letterplace implementation can only be used if the corresponding 

(multivariate) polynomial ring has an implementation in Singular:: 

 

sage: FreeAlgebra(FreeAlgebra(ZZ,2,'ab'), 2, 'x', implementation='letterplace') 

Traceback (most recent call last): 

... 

TypeError: The base ring Free Algebra on 2 generators (a, b) over Integer Ring is not a commutative ring 

""" 

 

#***************************************************************************** 

# Copyright (C) 2005 David Kohel <kohel@maths.usyd.edu> 

# Copyright (C) 2005,2006 William Stein <wstein@gmail.com> 

# Copyright (C) 2011 Simon King <simon.king@uni-jena.de> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

 

from __future__ import absolute_import 

from six.moves import range 

from six import integer_types 

import six 

 

from sage.categories.rings import Rings 

 

from sage.monoids.free_monoid import FreeMonoid 

from sage.monoids.free_monoid_element import FreeMonoidElement 

 

from sage.algebras.free_algebra_element import FreeAlgebraElement 

 

from sage.structure.factory import UniqueFactory 

from sage.misc.cachefunc import cached_method 

from sage.all import PolynomialRing 

from sage.rings.ring import Algebra 

from sage.categories.algebras_with_basis import AlgebrasWithBasis 

from sage.combinat.free_module import CombinatorialFreeModule 

from sage.combinat.words.word import Word 

from sage.structure.category_object import normalize_names 

 

 

class FreeAlgebraFactory(UniqueFactory): 

""" 

A constructor of free algebras. 

 

See :mod:`~sage.algebras.free_algebra` for examples and corner cases. 

 

EXAMPLES:: 

 

sage: FreeAlgebra(GF(5),3,'x') 

Free Algebra on 3 generators (x0, x1, x2) over Finite Field of size 5 

sage: F.<x,y,z> = FreeAlgebra(GF(5),3) 

sage: (x+y+z)^2 

x^2 + x*y + x*z + y*x + y^2 + y*z + z*x + z*y + z^2 

sage: FreeAlgebra(GF(5),3, 'xx, zba, Y') 

Free Algebra on 3 generators (xx, zba, Y) over Finite Field of size 5 

sage: FreeAlgebra(GF(5),3, 'abc') 

Free Algebra on 3 generators (a, b, c) over Finite Field of size 5 

sage: FreeAlgebra(GF(5),1, 'z') 

Free Algebra on 1 generators (z,) over Finite Field of size 5 

sage: FreeAlgebra(GF(5),1, ['alpha']) 

Free Algebra on 1 generators (alpha,) over Finite Field of size 5 

sage: FreeAlgebra(FreeAlgebra(ZZ,1,'a'), 2, 'x') 

Free Algebra on 2 generators (x0, x1) over Free Algebra on 1 generators (a,) over Integer Ring 

 

Free algebras are globally unique:: 

 

sage: F = FreeAlgebra(ZZ,3,'x,y,z') 

sage: G = FreeAlgebra(ZZ,3,'x,y,z') 

sage: F is G 

True 

sage: F.<x,y,z> = FreeAlgebra(GF(5),3) # indirect doctest 

sage: F is loads(dumps(F)) 

True 

sage: F is FreeAlgebra(GF(5),['x','y','z']) 

True 

sage: copy(F) is F is loads(dumps(F)) 

True 

sage: TestSuite(F).run() 

 

By :trac:`7797`, we provide a different implementation of free 

algebras, based on Singular's "letterplace rings". Our letterplace 

wrapper allows for chosing positive integral degree weights for the 

generators of the free algebra. However, only (weighted) homogenous 

elements are supported. Of course, isomorphic algebras in different 

implementations are not identical:: 

 

sage: G = FreeAlgebra(GF(5),['x','y','z'], implementation='letterplace') 

sage: F == G 

False 

sage: G is FreeAlgebra(GF(5),['x','y','z'], implementation='letterplace') 

True 

sage: copy(G) is G is loads(dumps(G)) 

True 

sage: TestSuite(G).run() 

 

:: 

 

sage: H = FreeAlgebra(GF(5),['x','y','z'], implementation='letterplace', degrees=[1,2,3]) 

sage: F != H != G 

True 

sage: H is FreeAlgebra(GF(5),['x','y','z'], implementation='letterplace', degrees=[1,2,3]) 

True 

sage: copy(H) is H is loads(dumps(H)) 

True 

sage: TestSuite(H).run() 

 

Free algebras commute with their base ring. 

:: 

 

sage: K.<a,b> = FreeAlgebra(QQ,2) 

sage: K.is_commutative() 

False 

sage: L.<c> = FreeAlgebra(K,1) 

sage: L.is_commutative() 

False 

sage: s = a*b^2 * c^3; s 

a*b^2*c^3 

sage: parent(s) 

Free Algebra on 1 generators (c,) over Free Algebra on 2 generators (a, b) over Rational Field 

sage: c^3 * a * b^2 

a*b^2*c^3 

""" 

def create_key(self, base_ring, arg1=None, arg2=None, 

sparse=None, order='degrevlex', 

names=None, name=None, 

implementation=None, degrees=None): 

""" 

Create the key under which a free algebra is stored. 

 

TESTS:: 

 

sage: FreeAlgebra.create_key(GF(5),['x','y','z']) 

(Finite Field of size 5, ('x', 'y', 'z')) 

sage: FreeAlgebra.create_key(GF(5),['x','y','z'],3) 

(Finite Field of size 5, ('x', 'y', 'z')) 

sage: FreeAlgebra.create_key(GF(5),3,'xyz') 

(Finite Field of size 5, ('x', 'y', 'z')) 

sage: FreeAlgebra.create_key(GF(5),['x','y','z'], implementation='letterplace') 

(Multivariate Polynomial Ring in x, y, z over Finite Field of size 5,) 

sage: FreeAlgebra.create_key(GF(5),['x','y','z'],3, implementation='letterplace') 

(Multivariate Polynomial Ring in x, y, z over Finite Field of size 5,) 

sage: FreeAlgebra.create_key(GF(5),3,'xyz', implementation='letterplace') 

(Multivariate Polynomial Ring in x, y, z over Finite Field of size 5,) 

sage: FreeAlgebra.create_key(GF(5),3,'xyz', implementation='letterplace', degrees=[1,2,3]) 

((1, 2, 3), Multivariate Polynomial Ring in x, y, z, x_ over Finite Field of size 5) 

 

""" 

if arg1 is None and arg2 is None and names is None: 

# this is used for pickling 

if degrees is None: 

return (base_ring,) 

return tuple(degrees),base_ring 

# test if we can use libSingular/letterplace 

if implementation == "letterplace": 

args = [arg for arg in (arg1, arg2) if arg is not None] 

kwds = dict(sparse=sparse, order=order, implementation="singular") 

if name is not None: 

kwds["name"] = name 

if names is not None: 

kwds["names"] = names 

PolRing = PolynomialRing(base_ring, *args, **kwds) 

if degrees is None: 

return (PolRing,) 

from sage.all import TermOrder 

T = PolRing.term_order() + TermOrder('lex',1) 

varnames = list(PolRing.variable_names()) 

newname = 'x' 

while newname in varnames: 

newname += '_' 

varnames.append(newname) 

R = PolynomialRing( 

PolRing.base(), varnames, 

sparse=sparse, order=T) 

return tuple(degrees), R 

# normalise the generator names 

from sage.all import Integer 

if isinstance(arg1, (Integer,) + integer_types): 

arg1, arg2 = arg2, arg1 

if not names is None: 

arg1 = names 

elif not name is None: 

arg1 = name 

if arg2 is None: 

arg2 = len(arg1) 

names = normalize_names(arg2, arg1) 

return base_ring, names 

 

def create_object(self, version, key): 

""" 

Construct the free algebra that belongs to a unique key. 

 

NOTE: 

 

Of course, that method should not be called directly, 

since it does not use the cache of free algebras. 

 

TESTS:: 

 

sage: FreeAlgebra.create_object('4.7.1', (QQ['x','y'],)) 

Free Associative Unital Algebra on 2 generators (x, y) over Rational Field 

sage: FreeAlgebra.create_object('4.7.1', (QQ['x','y'],)) is FreeAlgebra(QQ,['x','y']) 

False 

 

""" 

if len(key) == 1: 

from sage.algebras.letterplace.free_algebra_letterplace import FreeAlgebra_letterplace 

return FreeAlgebra_letterplace(key[0]) 

if isinstance(key[0], tuple): 

from sage.algebras.letterplace.free_algebra_letterplace import FreeAlgebra_letterplace 

return FreeAlgebra_letterplace(key[1], degrees=key[0]) 

return FreeAlgebra_generic(key[0], len(key[1]), key[1]) 

 

FreeAlgebra = FreeAlgebraFactory('FreeAlgebra') 

 

 

def is_FreeAlgebra(x): 

""" 

Return True if x is a free algebra; otherwise, return False. 

 

EXAMPLES:: 

 

sage: from sage.algebras.free_algebra import is_FreeAlgebra 

sage: is_FreeAlgebra(5) 

False 

sage: is_FreeAlgebra(ZZ) 

False 

sage: is_FreeAlgebra(FreeAlgebra(ZZ,100,'x')) 

True 

sage: is_FreeAlgebra(FreeAlgebra(ZZ,10,'x',implementation='letterplace')) 

True 

sage: is_FreeAlgebra(FreeAlgebra(ZZ,10,'x',implementation='letterplace', degrees=list(range(1,11)))) 

True 

 

""" 

from sage.algebras.letterplace.free_algebra_letterplace import FreeAlgebra_letterplace 

return isinstance(x, (FreeAlgebra_generic,FreeAlgebra_letterplace)) 

 

 

class FreeAlgebra_generic(CombinatorialFreeModule, Algebra): 

""" 

The free algebra on `n` generators over a base ring. 

 

INPUT: 

 

- ``R`` -- a ring 

- ``n`` -- an integer 

- ``names`` -- the generator names 

 

EXAMPLES:: 

 

sage: F.<x,y,z> = FreeAlgebra(QQ, 3); F 

Free Algebra on 3 generators (x, y, z) over Rational Field 

sage: mul(F.gens()) 

x*y*z 

sage: mul([ F.gen(i%3) for i in range(12) ]) 

x*y*z*x*y*z*x*y*z*x*y*z 

sage: mul([ F.gen(i%3) for i in range(12) ]) + mul([ F.gen(i%2) for i in range(12) ]) 

x*y*x*y*x*y*x*y*x*y*x*y + x*y*z*x*y*z*x*y*z*x*y*z 

sage: (2 + x*z + x^2)^2 + (x - y)^2 

4 + 5*x^2 - x*y + 4*x*z - y*x + y^2 + x^4 + x^3*z + x*z*x^2 + x*z*x*z 

 

TESTS: 

 

Free algebras commute with their base ring. 

:: 

 

sage: K.<a,b> = FreeAlgebra(QQ) 

sage: K.is_commutative() 

False 

sage: L.<c,d> = FreeAlgebra(K) 

sage: L.is_commutative() 

False 

sage: s = a*b^2 * c^3; s 

a*b^2*c^3 

sage: parent(s) 

Free Algebra on 2 generators (c, d) over Free Algebra on 2 generators (a, b) over Rational Field 

sage: c^3 * a * b^2 

a*b^2*c^3 

 

""" 

Element = FreeAlgebraElement 

def __init__(self, R, n, names): 

""" 

The free algebra on `n` generators over a base ring. 

 

EXAMPLES:: 

 

sage: F.<x,y,z> = FreeAlgebra(QQ, 3); F # indirect doctet 

Free Algebra on 3 generators (x, y, z) over Rational Field 

 

TESTS: 

 

Note that the following is *not* the recommended way to create 

a free algebra:: 

 

sage: from sage.algebras.free_algebra import FreeAlgebra_generic 

sage: FreeAlgebra_generic(ZZ, 3, 'abc') 

Free Algebra on 3 generators (a, b, c) over Integer Ring 

""" 

if R not in Rings(): 

raise TypeError("Argument R must be a ring.") 

self.__ngens = n 

indices = FreeMonoid(n, names=names) 

cat = AlgebrasWithBasis(R) 

CombinatorialFreeModule.__init__(self, R, indices, prefix='F', 

category=cat) 

self._assign_names(indices.variable_names()) 

 

def one_basis(self): 

""" 

Return the index of the basis element `1`. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(QQ, 2, 'x,y') 

sage: F.one_basis() 

1 

sage: F.one_basis().parent() 

Free monoid on 2 generators (x, y) 

""" 

return self._indices.one() 

 

def is_field(self, proof=True): 

""" 

Return True if this Free Algebra is a field, which is only if the 

base ring is a field and there are no generators 

 

EXAMPLES:: 

 

sage: A = FreeAlgebra(QQ,0,'') 

sage: A.is_field() 

True 

sage: A = FreeAlgebra(QQ,1,'x') 

sage: A.is_field() 

False 

""" 

if self.__ngens == 0: 

return self.base_ring().is_field(proof) 

return False 

 

def is_commutative(self): 

""" 

Return True if this free algebra is commutative. 

 

EXAMPLES:: 

 

sage: R.<x> = FreeAlgebra(QQ,1) 

sage: R.is_commutative() 

True 

sage: R.<x,y> = FreeAlgebra(QQ,2) 

sage: R.is_commutative() 

False 

""" 

return self.__ngens <= 1 and self.base_ring().is_commutative() 

 

def __eq__(self, other): 

""" 

Two free algebras are considered the same if they have the same 

base ring, number of generators and variable names, and the same 

implementation. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(QQ,3,'x') 

sage: F == FreeAlgebra(QQ,3,'x') 

True 

sage: F is FreeAlgebra(QQ,3,'x') 

True 

sage: F == FreeAlgebra(ZZ,3,'x') 

False 

sage: F == FreeAlgebra(QQ,4,'x') 

False 

sage: F == FreeAlgebra(QQ,3,'y') 

False 

 

Note that since :trac:`7797` there is a different 

implementation of free algebras. Two corresponding free 

algebras in different implementations are not equal, but there 

is a coercion. 

""" 

if not isinstance(other, FreeAlgebra_generic): 

return False 

if self.base_ring() != other.base_ring(): 

return False 

if self.__ngens != other.ngens(): 

return False 

if self.variable_names() != other.variable_names(): 

return False 

return True 

 

def _repr_(self): 

""" 

Text representation of this free algebra. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(QQ,3,'x') 

sage: F # indirect doctest 

Free Algebra on 3 generators (x0, x1, x2) over Rational Field 

sage: F.rename('QQ<<x0,x1,x2>>') 

sage: F #indirect doctest 

QQ<<x0,x1,x2>> 

sage: FreeAlgebra(ZZ,1,['a']) 

Free Algebra on 1 generators (a,) over Integer Ring 

""" 

return "Free Algebra on {} generators {} over {}".format( 

self.__ngens, self.gens(), self.base_ring()) 

 

def _latex_(self): 

r""" 

Return a latex representation of ``self``. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(QQ,3,'x') 

sage: latex(F) 

\Bold{Q}\langle x_{0}, x_{1}, x_{2}\rangle 

sage: F = FreeAlgebra(ZZ['q'], 3, 'a,b,c') 

sage: latex(F) 

\Bold{Z}[q]\langle a, b, c\rangle 

""" 

from sage.misc.latex import latex 

return "{}\\langle {}\\rangle".format(latex(self.base_ring()), 

', '.join(self.latex_variable_names())) 

 

def _element_constructor_(self, x): 

""" 

Convert ``x`` into ``self``. 

 

EXAMPLES:: 

 

sage: R.<x,y> = FreeAlgebra(QQ,2) 

sage: R(3) # indirect doctest 

3 

 

TESTS:: 

 

sage: F.<x,y,z> = FreeAlgebra(GF(5),3) 

sage: L.<x,y,z> = FreeAlgebra(ZZ,3,implementation='letterplace') 

sage: F(x) # indirect doctest 

x 

sage: F.1*L.2 

y*z 

sage: (F.1*L.2).parent() is F 

True 

 

:: 

 

sage: K.<z> = GF(25) 

sage: F.<a,b,c> = FreeAlgebra(K,3) 

sage: L.<a,b,c> = FreeAlgebra(K,3, implementation='letterplace') 

sage: F.1+(z+1)*L.2 

b + (z+1)*c 

 

Check that :trac:`15169` is fixed:: 

 

sage: A.<x> = FreeAlgebra(CC) 

sage: A(2) 

2.00000000000000 

 

We check that the string coercions work correctly over 

inexact fields:: 

 

sage: F.<x,y> = FreeAlgebra(CC) 

sage: F('2') 

2.00000000000000 

sage: F('x') 

1.00000000000000*x 

 

Check that it also converts factorizations:: 

 

sage: f = Factorization([(x,2),(y,3)]); f 

1.00000000000000*x^2 * 1.00000000000000*y^3 

sage: F(f) 

1.00000000000000*x^2*y^3 

""" 

if isinstance(x, FreeAlgebraElement): 

P = x.parent() 

if P is self: 

return x 

if P is not self.base_ring(): 

return self.element_class(self, x) 

elif hasattr(x,'letterplace_polynomial'): 

P = x.parent() 

if self.has_coerce_map_from(P): # letterplace versus generic 

ngens = P.ngens() 

M = self._indices 

def exp_to_monomial(T): 

out = [] 

for i in range(len(T)): 

if T[i]: 

out.append((i%ngens,T[i])) 

return M(out) 

return self.element_class(self, {exp_to_monomial(T):c for T,c in six.iteritems(x.letterplace_polynomial().dict())}) 

# ok, not a free algebra element (or should not be viewed as one). 

if isinstance(x, six.string_types): 

from sage.all import sage_eval 

G = self.gens() 

d = {str(v): G[i] for i,v in enumerate(self.variable_names())} 

return self(sage_eval(x, locals=d)) 

R = self.base_ring() 

# coercion from free monoid 

if isinstance(x, FreeMonoidElement) and x.parent() is self._indices: 

return self.element_class(self, {x: R.one()}) 

# coercion from the PBW basis 

if isinstance(x, PBWBasisOfFreeAlgebra.Element) \ 

and self.has_coerce_map_from(x.parent()._alg): 

return self(x.parent().expansion(x)) 

 

# Check if it's a factorization 

from sage.structure.factorization import Factorization 

if isinstance(x, Factorization): 

return self.prod(f**i for f,i in x) 

 

# coercion via base ring 

x = R(x) 

if x == 0: 

return self.element_class(self, {}) 

return self.element_class(self, {self.one_basis(): x}) 

 

def _coerce_map_from_(self, R): 

""" 

Return ``True`` if there is a coercion from ``R`` into ``self`` and 

``False`` otherwise. The things that coerce into ``self`` are: 

 

- This free algebra. 

 

- Anything with a coercion into ``self.monoid()``. 

 

- Free algebras in the same variables over a base with a coercion 

map into ``self.base_ring()``. 

 

- The underlying monoid. 

 

- The PBW basis of ``self``. 

 

- Anything with a coercion into ``self.base_ring()``. 

 

TESTS:: 

 

sage: F = FreeAlgebra(ZZ, 3, 'x,y,z') 

sage: G = FreeAlgebra(QQ, 3, 'x,y,z') 

sage: H = FreeAlgebra(ZZ, 1, 'y') 

sage: F._coerce_map_from_(G) 

False 

sage: G._coerce_map_from_(F) 

True 

sage: F._coerce_map_from_(H) 

False 

sage: F._coerce_map_from_(QQ) 

False 

sage: G._coerce_map_from_(QQ) 

True 

sage: F._coerce_map_from_(G.monoid()) 

True 

sage: F._coerce_map_from_(F.pbw_basis()) 

True 

sage: F.has_coerce_map_from(PolynomialRing(ZZ, 3, 'x,y,z')) 

False 

 

sage: K.<z> = GF(25) 

sage: F.<a,b,c> = FreeAlgebra(K,3) 

sage: F._coerce_map_from_(ZZ) 

True 

sage: F._coerce_map_from_(QQ) 

False 

sage: F._coerce_map_from_(F.monoid()) 

True 

sage: F._coerce_map_from_(F.pbw_basis()) 

True 

sage: G = FreeAlgebra(ZZ, 3, 'a,b,c') 

sage: F._coerce_map_from_(G) 

True 

sage: G._coerce_map_from_(F) 

False 

sage: L.<a,b,c> = FreeAlgebra(K,3, implementation='letterplace') 

sage: F.1 + (z+1) * L.2 

b + (z+1)*c 

""" 

if self._indices.has_coerce_map_from(R): 

return True 

 

# free algebras in the same variable over any base that coerces in: 

if is_FreeAlgebra(R): 

if R.variable_names() == self.variable_names(): 

return self.base_ring().has_coerce_map_from(R.base_ring()) 

if isinstance(R, PBWBasisOfFreeAlgebra): 

return self.has_coerce_map_from(R._alg) 

 

return self.base_ring().has_coerce_map_from(R) 

 

def gen(self, i): 

""" 

The ``i``-th generator of the algebra. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(ZZ,3,'x,y,z') 

sage: F.gen(0) 

x 

""" 

if i < 0 or not i < self.__ngens: 

raise IndexError("Argument i (= {}) must be between 0 and {}.".format(i, self.__ngens-1)) 

R = self.base_ring() 

F = self._indices 

return self.element_class(self, {F.gen(i): R.one()}) 

 

@cached_method 

def algebra_generators(self): 

""" 

Return the algebra generators of ``self``. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(ZZ,3,'x,y,z') 

sage: F.algebra_generators() 

Finite family {'y': y, 'x': x, 'z': z} 

""" 

ret = {} 

for i in range(self.__ngens): 

x = self.gen(i) 

ret[str(x)] = x 

from sage.sets.family import Family 

return Family(self.variable_names(), lambda i: ret[i]) 

 

@cached_method 

def gens(self): 

""" 

Return the generators of ``self``. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(ZZ,3,'x,y,z') 

sage: F.gens() 

(x, y, z) 

""" 

return tuple(self.gen(i) for i in range(self.__ngens)) 

 

def product_on_basis(self, x, y): 

""" 

Return the product of the basis elements indexed by ``x`` and ``y``. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(ZZ,3,'x,y,z') 

sage: I = F.basis().keys() 

sage: x,y,z = I.gens() 

sage: F.product_on_basis(x*y, z*y) 

x*y*z*y 

""" 

return self.monomial(x * y) 

 

def quotient(self, mons, mats=None, names=None): 

""" 

Return a quotient algebra. 

 

The quotient algebra is defined via the action of a free algebra 

`A` on a (finitely generated) free module. The input for the quotient 

algebra is a list of monomials (in the underlying monoid for `A`) 

which form a free basis for the module of `A`, and a list of 

matrices, which give the action of the free generators of `A` on this 

monomial basis. 

 

EXAMPLES: 

 

Here is the quaternion algebra defined in terms of three generators:: 

 

sage: n = 3 

sage: A = FreeAlgebra(QQ,n,'i') 

sage: F = A.monoid() 

sage: i, j, k = F.gens() 

sage: mons = [ F(1), i, j, k ] 

sage: M = MatrixSpace(QQ,4) 

sage: mats = [M([0,1,0,0, -1,0,0,0, 0,0,0,-1, 0,0,1,0]), M([0,0,1,0, 0,0,0,1, -1,0,0,0, 0,-1,0,0]), M([0,0,0,1, 0,0,-1,0, 0,1,0,0, -1,0,0,0]) ] 

sage: H.<i,j,k> = A.quotient(mons, mats); H 

Free algebra quotient on 3 generators ('i', 'j', 'k') and dimension 4 over Rational Field 

""" 

if mats is None: 

return super(FreeAlgebra_generic, self).quotient(mons, names) 

from . import free_algebra_quotient 

return free_algebra_quotient.FreeAlgebraQuotient(self, mons, mats, names) 

 

quo = quotient 

 

def ngens(self): 

""" 

The number of generators of the algebra. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(ZZ,3,'x,y,z') 

sage: F.ngens() 

3 

""" 

return self.__ngens 

 

def monoid(self): 

""" 

The free monoid of generators of the algebra. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(ZZ,3,'x,y,z') 

sage: F.monoid() 

Free monoid on 3 generators (x, y, z) 

""" 

return self._indices 

 

def g_algebra(self, relations, names=None, order='degrevlex', check=True): 

""" 

The `G`-Algebra derived from this algebra by relations. 

By default is assumed, that two variables commute. 

 

.. TODO:: 

 

- Coercion doesn't work yet, there is some cheating about assumptions 

- The optional argument ``check`` controls checking the degeneracy 

conditions. Furthermore, the default values interfere with 

non-degeneracy conditions. 

 

EXAMPLES:: 

 

sage: A.<x,y,z> = FreeAlgebra(QQ,3) 

sage: G = A.g_algebra({y*x: -x*y}) 

sage: (x,y,z) = G.gens() 

sage: x*y 

x*y 

sage: y*x 

-x*y 

sage: z*x 

x*z 

sage: (x,y,z) = A.gens() 

sage: G = A.g_algebra({y*x: -x*y+1}) 

sage: (x,y,z) = G.gens() 

sage: y*x 

-x*y + 1 

sage: (x,y,z) = A.gens() 

sage: G = A.g_algebra({y*x: -x*y+z}) 

sage: (x,y,z) = G.gens() 

sage: y*x 

-x*y + z 

""" 

from sage.matrix.constructor import Matrix 

 

base_ring = self.base_ring() 

n = self.__ngens 

cmat = Matrix(base_ring, n) 

dmat = Matrix(self, n) 

for i in range(n): 

for j in range(i + 1, n): 

cmat[i,j] = 1 

for (to_commute,commuted) in six.iteritems(relations): 

#This is dirty, coercion is broken 

assert isinstance(to_commute, FreeAlgebraElement), to_commute.__class__ 

assert isinstance(commuted, FreeAlgebraElement), commuted 

((v1,e1),(v2,e2)) = list(list(to_commute)[0][0]) 

assert e1 == 1 

assert e2 == 1 

assert v1 > v2 

c_coef = None 

d_poly = None 

for (m,c) in commuted: 

if list(m) == [(v2,1),(v1,1)]: 

c_coef = c 

#buggy coercion workaround 

d_poly = commuted - self(c) * self(m) 

break 

assert not c_coef is None,list(m) 

v2_ind = self.gens().index(v2) 

v1_ind = self.gens().index(v1) 

cmat[v2_ind,v1_ind] = c_coef 

if d_poly: 

dmat[v2_ind,v1_ind] = d_poly 

from sage.rings.polynomial.plural import g_Algebra 

return g_Algebra(base_ring, cmat, dmat, names = names or self.variable_names(), 

order=order, check=check) 

 

def poincare_birkhoff_witt_basis(self): 

""" 

Return the Poincaré-Birkhoff-Witt (PBW) basis of ``self``. 

 

EXAMPLES:: 

 

sage: F.<x,y> = FreeAlgebra(QQ, 2) 

sage: F.poincare_birkhoff_witt_basis() 

The Poincare-Birkhoff-Witt basis of Free Algebra on 2 generators (x, y) over Rational Field 

""" 

return PBWBasisOfFreeAlgebra(self) 

 

pbw_basis = poincare_birkhoff_witt_basis 

 

def pbw_element(self, elt): 

""" 

Return the element ``elt`` in the Poincaré-Birkhoff-Witt basis. 

 

EXAMPLES:: 

 

sage: F.<x,y> = FreeAlgebra(QQ, 2) 

sage: F.pbw_element(x*y - y*x + 2) 

2*PBW[1] + PBW[x*y] 

sage: F.pbw_element(F.one()) 

PBW[1] 

sage: F.pbw_element(x*y*x + x^3*y) 

PBW[x*y]*PBW[x] + PBW[y]*PBW[x]^2 + PBW[x^3*y] 

+ 3*PBW[x^2*y]*PBW[x] + 3*PBW[x*y]*PBW[x]^2 + PBW[y]*PBW[x]^3 

""" 

PBW = self.pbw_basis() 

if elt == self.zero(): 

return PBW.zero() 

 

l = {} 

while elt: # != 0 

lst = list(elt) 

support = [i[0].to_word() for i in lst] 

min_elt = support[0] 

for word in support[1:len(support)-1]: 

if min_elt.lex_less(word): 

min_elt = word 

coeff = lst[support.index(min_elt)][1] 

min_elt = min_elt.to_monoid_element() 

l[min_elt] = l.get(min_elt, 0) + coeff 

elt = elt - coeff * self.lie_polynomial(min_elt) 

return PBW.sum_of_terms([(k, v) for k,v in l.items() if v != 0], distinct=True) 

 

def lie_polynomial(self, w): 

""" 

Return the Lie polynomial associated to the Lyndon word ``w``. If 

``w`` is not Lyndon, then return the product of Lie polynomials of 

the Lyndon factorization of ``w``. 

 

Given a Lyndon word `w`, the Lie polynomial `L_w` is defined 

recursively by `L_w = [L_u, L_v]`, where `w = uv` is the 

:meth:`standard factorization 

<sage.combinat.words.finite_word.FiniteWord_class.standard_factorization>` 

of `w`, and `L_w = w` when `w` is a single letter. 

 

INPUT: 

 

- ``w`` -- a word or an element of the free monoid 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(QQ, 3, 'x,y,z') 

sage: M.<x,y,z> = FreeMonoid(3) 

sage: F.lie_polynomial(x*y) 

x*y - y*x 

sage: F.lie_polynomial(y*x) 

y*x 

sage: F.lie_polynomial(x^2*y*x) 

x^2*y*x - 2*x*y*x^2 + y*x^3 

sage: F.lie_polynomial(y*z*x*z*x*z) 

y*z*x*z*x*z - y*z*x*z^2*x - y*z^2*x^2*z + y*z^2*x*z*x 

- z*y*x*z*x*z + z*y*x*z^2*x + z*y*z*x^2*z - z*y*z*x*z*x 

 

TESTS: 

 

We test some corner cases and alternative inputs:: 

 

sage: F = FreeAlgebra(QQ, 3, 'x,y,z') 

sage: M.<x,y,z> = FreeMonoid(3) 

sage: F.lie_polynomial(Word('xy')) 

x*y - y*x 

sage: F.lie_polynomial('xy') 

x*y - y*x 

sage: F.lie_polynomial(M.one()) 

1 

sage: F.lie_polynomial(Word([])) 

1 

sage: F.lie_polynomial('') 

1 

 

We check that :trac:`22251` is fixed:: 

 

sage: F.lie_polynomial(x*y*z) 

x*y*z - x*z*y - y*z*x + z*y*x 

""" 

if not w: 

return self.one() 

M = self._indices 

 

if len(w) == 1: 

return self(M(w)) 

 

ret = self.one() 

# We have to be careful about order here. 

# Since the Lyndon factors appear from left to right 

# we must multiply from left to right as well. 

for factor in Word(w).lyndon_factorization(): 

if len(factor) == 1: 

ret = ret * self(M(factor)) 

continue 

x,y = factor.standard_factorization() 

x = self.lie_polynomial(M(x)) 

y = self.lie_polynomial(M(y)) 

ret = ret * (x*y - y*x) 

return ret 

 

 

class PBWBasisOfFreeAlgebra(CombinatorialFreeModule): 

""" 

The Poincaré-Birkhoff-Witt basis of the free algebra. 

 

EXAMPLES:: 

 

sage: F.<x,y> = FreeAlgebra(QQ, 2) 

sage: PBW = F.pbw_basis() 

sage: px, py = PBW.gens() 

sage: px * py 

PBW[x*y] + PBW[y]*PBW[x] 

sage: py * px 

PBW[y]*PBW[x] 

sage: px * py^3 * px - 2*px * py 

-2*PBW[x*y] - 2*PBW[y]*PBW[x] + PBW[x*y^3]*PBW[x] 

+ 3*PBW[y]*PBW[x*y^2]*PBW[x] + 3*PBW[y]^2*PBW[x*y]*PBW[x] 

+ PBW[y]^3*PBW[x]^2 

 

We can convert between the two bases:: 

 

sage: p = PBW(x*y - y*x + 2); p 

2*PBW[1] + PBW[x*y] 

sage: F(p) 

2 + x*y - y*x 

sage: f = F.pbw_element(x*y*x + x^3*y + x + 3) 

sage: F(PBW(f)) == f 

True 

sage: p = px*py + py^4*px^2 

sage: F(p) 

x*y + y^4*x^2 

sage: PBW(F(p)) == p 

True 

 

Note that multiplication in the PBW basis agrees with multiplication 

as monomials:: 

 

sage: F(px * py^3 * px - 2*px * py) == x*y^3*x - 2*x*y 

True 

 

We verify Examples 1 and 2 in [MR1989]_:: 

 

sage: F.<x,y,z> = FreeAlgebra(QQ) 

sage: PBW = F.pbw_basis() 

sage: PBW(x*y*z) 

PBW[x*y*z] + PBW[x*z*y] + PBW[y]*PBW[x*z] + PBW[y*z]*PBW[x] 

+ PBW[z]*PBW[x*y] + PBW[z]*PBW[y]*PBW[x] 

sage: PBW(x*y*y*x) 

PBW[x*y^2]*PBW[x] + 2*PBW[y]*PBW[x*y]*PBW[x] + PBW[y]^2*PBW[x]^2 

 

TESTS: 

 

Check that going between the two bases is the identity:: 

 

sage: F = FreeAlgebra(QQ, 2, 'x,y') 

sage: PBW = F.pbw_basis() 

sage: M = F.monoid() 

sage: L = [j.to_monoid_element() for i in range(6) for j in Words('xy', i)] 

sage: all(PBW(F(PBW(m))) == PBW(m) for m in L) 

True 

sage: all(F(PBW(F(m))) == F(m) for m in L) 

True 

""" 

@staticmethod 

def __classcall_private__(cls, R, n=None, names=None): 

""" 

Normalize input to ensure a unique representation. 

 

EXAMPLES:: 

 

sage: from sage.algebras.free_algebra import PBWBasisOfFreeAlgebra 

sage: PBW1 = FreeAlgebra(QQ, 2, 'x,y').pbw_basis() 

sage: PBW2.<x,y> = PBWBasisOfFreeAlgebra(QQ) 

sage: PBW3 = PBWBasisOfFreeAlgebra(QQ, 2, ['x','y']) 

sage: PBW1 is PBW2 and PBW2 is PBW3 

True 

""" 

if n is None and names is None: 

if not isinstance(R, FreeAlgebra_generic): 

raise ValueError("{} is not a free algebra".format(R)) 

alg = R 

else: 

if n is None: 

n = len(names) 

alg = FreeAlgebra(R, n, names) 

return super(PBWBasisOfFreeAlgebra, cls).__classcall__(cls, alg) 

 

def __init__(self, alg): 

""" 

Initialize ``self``. 

 

EXAMPLES:: 

 

sage: PBW = FreeAlgebra(QQ, 2, 'x,y').pbw_basis() 

sage: TestSuite(PBW).run() 

""" 

R = alg.base_ring() 

self._alg = alg 

category = AlgebrasWithBasis(R) 

CombinatorialFreeModule.__init__(self, R, alg.monoid(), prefix='PBW', 

category=category) 

self._assign_names(alg.variable_names()) 

 

def _repr_(self): 

""" 

Return a string representation of ``self``. 

 

EXAMPLES:: 

 

sage: FreeAlgebra(QQ, 2, 'x,y').pbw_basis() 

The Poincare-Birkhoff-Witt basis of Free Algebra on 2 generators (x, y) over Rational Field 

""" 

return "The Poincare-Birkhoff-Witt basis of {}".format(self._alg) 

 

def _repr_term(self, w): 

""" 

Return a representation of term indexed by ``w``. 

 

EXAMPLES:: 

 

sage: PBW = FreeAlgebra(QQ, 2, 'x,y').pbw_basis() 

sage: x,y = PBW.gens() 

sage: x*y # indirect doctest 

PBW[x*y] + PBW[y]*PBW[x] 

sage: y*x 

PBW[y]*PBW[x] 

sage: x^3 

PBW[x]^3 

sage: PBW.one() 

PBW[1] 

sage: 3*PBW.one() 

3*PBW[1] 

""" 

if len(w) == 0: 

return super(PBWBasisOfFreeAlgebra, self)._repr_term(w) 

ret = '' 

p = 1 

cur = None 

for x in w.to_word().lyndon_factorization(): 

if x == cur: 

p += 1 

else: 

if len(ret) != 0: 

if p != 1: 

ret += "^{}".format(p) 

ret += "*" 

ret += super(PBWBasisOfFreeAlgebra, self)._repr_term(x.to_monoid_element()) 

cur = x 

p = 1 

if p != 1: 

ret += "^{}".format(p) 

return ret 

 

def _element_constructor_(self, x): 

""" 

Convert ``x`` into ``self``. 

 

EXAMPLES:: 

 

sage: F.<x,y> = FreeAlgebra(QQ, 2) 

sage: R = F.pbw_basis() 

sage: R(3) 

3*PBW[1] 

sage: R(x*y) 

PBW[x*y] + PBW[y]*PBW[x] 

""" 

if isinstance(x, FreeAlgebraElement): 

return self._alg.pbw_element(self._alg(x)) 

return CombinatorialFreeModule._element_constructor_(self, x) 

 

def _coerce_map_from_(self, R): 

""" 

Return ``True`` if there is a coercion from ``R`` into ``self`` and 

``False`` otherwise. The things that coerce into ``self`` are: 

 

- Anything that coerces into the associated free algebra of ``self`` 

 

TESTS:: 

 

sage: F = FreeAlgebra(ZZ, 3, 'x,y,z').pbw_basis() 

sage: G = FreeAlgebra(QQ, 3, 'x,y,z').pbw_basis() 

sage: H = FreeAlgebra(ZZ, 1, 'y').pbw_basis() 

sage: F._coerce_map_from_(G) 

False 

sage: G._coerce_map_from_(F) 

True 

sage: F._coerce_map_from_(H) 

False 

sage: F._coerce_map_from_(QQ) 

False 

sage: G._coerce_map_from_(QQ) 

True 

sage: F._coerce_map_from_(G._alg.monoid()) 

True 

sage: F.has_coerce_map_from(PolynomialRing(ZZ, 3, 'x,y,z')) 

False 

sage: F.has_coerce_map_from(FreeAlgebra(ZZ, 3, 'x,y,z')) 

True 

""" 

return self._alg.has_coerce_map_from(R) 

 

def one_basis(self): 

""" 

Return the index of the basis element for `1`. 

 

EXAMPLES:: 

 

sage: PBW = FreeAlgebra(QQ, 2, 'x,y').pbw_basis() 

sage: PBW.one_basis() 

1 

sage: PBW.one_basis().parent() 

Free monoid on 2 generators (x, y) 

""" 

return self._indices.one() 

 

def algebra_generators(self): 

""" 

Return the generators of ``self`` as an algebra. 

 

EXAMPLES:: 

 

sage: PBW = FreeAlgebra(QQ, 2, 'x,y').pbw_basis() 

sage: gens = PBW.algebra_generators(); gens 

(PBW[x], PBW[y]) 

sage: all(g.parent() is PBW for g in gens) 

True 

""" 

return tuple(self.monomial(x) for x in self._indices.gens()) 

 

gens = algebra_generators 

 

def gen(self, i): 

""" 

Return the ``i``-th generator of ``self``. 

 

EXAMPLES:: 

 

sage: PBW = FreeAlgebra(QQ, 2, 'x,y').pbw_basis() 

sage: PBW.gen(0) 

PBW[x] 

sage: PBW.gen(1) 

PBW[y] 

""" 

return self.algebra_generators()[i] 

 

def free_algebra(self): 

""" 

Return the associated free algebra of ``self``. 

 

EXAMPLES:: 

 

sage: PBW = FreeAlgebra(QQ, 2, 'x,y').pbw_basis() 

sage: PBW.free_algebra() 

Free Algebra on 2 generators (x, y) over Rational Field 

""" 

return self._alg 

 

def product(self, u, v): 

""" 

Return the product of two elements ``u`` and ``v``. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(QQ, 2, 'x,y') 

sage: PBW = F.pbw_basis() 

sage: x, y = PBW.gens() 

sage: PBW.product(x, y) 

PBW[x*y] + PBW[y]*PBW[x] 

sage: PBW.product(y, x) 

PBW[y]*PBW[x] 

sage: PBW.product(y^2*x, x*y*x) 

PBW[y]^2*PBW[x^2*y]*PBW[x] + 2*PBW[y]^2*PBW[x*y]*PBW[x]^2 + PBW[y]^3*PBW[x]^3 

 

TESTS: 

 

Check that multiplication agrees with the multiplication in the 

free algebra:: 

 

sage: F = FreeAlgebra(QQ, 2, 'x,y') 

sage: PBW = F.pbw_basis() 

sage: x, y = PBW.gens() 

sage: F(x*y) 

x*y 

sage: F(x*y*x) 

x*y*x 

sage: PBW(F(x)*F(y)*F(x)) == x*y*x 

True 

""" 

return self(self.expansion(u) * self.expansion(v)) 

 

def expansion(self, t): 

""" 

Return the expansion of the element ``t`` of the Poincaré-Birkhoff-Witt 

basis in the monomials of the free algebra. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(QQ, 2, 'x,y') 

sage: PBW = F.pbw_basis() 

sage: x,y = F.monoid().gens() 

sage: PBW.expansion(PBW(x*y)) 

x*y - y*x 

sage: PBW.expansion(PBW.one()) 

1 

sage: PBW.expansion(PBW(x*y*x) + 2*PBW(x) + 3) 

3 + 2*x + x*y*x - y*x^2 

 

TESTS: 

 

Check that we have the correct parent:: 

 

sage: PBW.expansion(PBW(x*y)).parent() is F 

True 

sage: PBW.expansion(PBW.one()).parent() is F 

True 

""" 

return sum([i[1] * self._alg.lie_polynomial(i[0]) for i in list(t)], 

self._alg.zero()) 

 

class Element(CombinatorialFreeModule.Element): 

def expand(self): 

""" 

Expand ``self`` in the monomials of the free algebra. 

 

EXAMPLES:: 

 

sage: F = FreeAlgebra(QQ, 2, 'x,y') 

sage: PBW = F.pbw_basis() 

sage: x,y = F.monoid().gens() 

sage: f = PBW(x^2*y) + PBW(x) + PBW(y^4*x) 

sage: f.expand() 

x + x^2*y - 2*x*y*x + y*x^2 + y^4*x 

""" 

return self.parent().expansion(self)