Module: sage.algebras.free_algebra_quotient
Free algebra quotients
TESTS:
sage: n = 2 sage: A = FreeAlgebra(QQ,n,'x') sage: F = A.monoid() sage: i, j = F.gens() sage: mons = [ F(1), i, j, i*j ] sage: r = len(mons) sage: M = MatrixSpace(QQ,r) 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]) ] sage: H2.<i,j> = A.quotient(mons,mats) sage: H2 == loads(dumps(H2)) True sage: i == loads(dumps(i)) True
Class: FreeAlgebraQuotient
self, A, mons, mats, names) |
Returns a quotient algebra 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.
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: H3.<i,j,k> = FreeAlgebraQuotient(A,mons,mats) sage: x = 1 + i + j + k sage: x 1 + i + j + k sage: x**128 -170141183460469231731687303715884105728 + 170141183460469231731687303715884105728*i + 170141183460469231731687303715884105728*j + 170141183460469231731687303715884105728*k
Same algebra defined in terms of two generators, with some penalty on already slow arithmetic.
sage: n = 2 sage: A = FreeAlgebra(QQ,n,'x') sage: F = A.monoid() sage: i, j = F.gens() sage: mons = [ F(1), i, j, i*j ] sage: r = len(mons) sage: M = MatrixSpace(QQ,r) 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]) ] sage: H2.<i,j> = A.quotient(mons,mats) sage: k = i*j sage: x = 1 + i + j + k sage: x 1 + i + j + i*j sage: x**128 -170141183460469231731687303715884105728 + 170141183460469231731687303715884105728*i + 170141183460469231731687303715884105728*j + 170141183460469231731687303715884105728*i*j
Functions: dimension,
free_algebra,
gen,
matrix_action,
module,
monoid,
monomial_basis,
ngens,
rank
self) |
The rank of the algebra (as a free module).
self) |
The free algebra generating the algebra.
self, i) |
The i-th generator of the algebra.
self) |
The free module of the algebra.
self) |
The free monoid of generators of the algebra.
self) |
The free monoid of generators of the algebra as elements of a free monoid.
self) |
The number of generators of the algebra.
self) |
The rank of the algebra (as a free module).
Special Functions: __call__,
__contains__,
__eq__,
__init__,
_coerce_impl,
_repr_
self, x) |
Return the coercion of x into this free algebra quotient.
The algebras that coerce into this quotient ring canonically, are:
* this quotient algebra * anything that coerces into the algebra of which this is the quotient