An implementation of finitely presented algebras into SageMath. Creates two classes, FinitelyPresentedAlgebra and FinitelyPresentedAlgebraElement, and provides functions that implements algorithms for finite dimensional representations of finitely presented algebras.

To start, open Sage in the same directory as the file finitely_presented_algebra.py. Then we can import the file.

sage: from finitely_presented_algebra import *

With this, we can now create our FinitelyPresentedAlgebra object. We do so with FinitelyPresentedAlgebra(field, relations, names). The input field must be a computable field, and relations and names may be given as a tuple, array, or comma delimited string.

sage: A = FinitelyPresentedAlgebra(QQ, 'b*a - a*b - 1', 'a, b')
sage: A
Finitely presented algebra over Rational Field with presentation <a, b | -1 - a*b + b*a>

Note that in this implementation variables must be injected before they can be used. We can also declare names in a similar way as to polynomial rings in Sage.

sage: R.<x,y> = FinitelyPresentedAlgebra(QQ, 'x*y + y*x')
sage: R
Finitely presented algebra over Rational Field with presentation <x, y | x*y + y*x>

We are also equipped with the class FinitelyPresentedAlgebraElement. An instance of this class can be created by calling the element constructor of FinitelyPresentedAlgebra in the usual way.

sage: R(x+ y)
x + y

When we have relations with two or fewer terms, we can reduce elements under the right conditions.

sage: R(2*x*y + y*x + 1)
1 + x*y

• lift()

Returns the value of self lifted to the base free algebra of its parent. The returned value will be a FreeAlgebraElement.

• is_constant()

Returns True if self is constant, and False otherwise. That is, whether self is contained in the base field of its parent.

• ngens()

Returns the number of generations of self.

• nrels()

Returns the number of relations of self.

• base_ring()

Returns the base field of self, which is field when self is initialized.

• base_field()

Same functionality as base_ring().

• free_algebra()

Returns the base free algebra of self, which is a FreeAlgebra object over its base field with generators matching the generators of self.

• gen(i)

Returns the i-th generator of self, as a FreeAlgebraElement.

• gens()

Returns the generators of self, as a tuple.

• rel(i)

Returns the i-th relation of self, as a FreeAlgebraElement.

• rels()

Returns the relations of self, as a tuple.

• one()

Returns the multiplicative identity of self, which is equal to the multiplicative identity of its base field.

• zero()

Returns the additive identity of self, which is equal to the additive identity of its base field.

• monoid()

Returns the free monoid on the generators of self.

• has_rep(n, restrict=None, force=False)

Returns True if there exists an n-dimensional representation of self, and False otherwise.

sage: R.<x,y> = FinitelyPresentedAlgebra(QQ, 'x*y + y*x')
sage: R.has_rep(2)
True
sage: A.<a,b> = FinitelyPresentedAlgebra(QQ, 'b*a - a*b - 1')
sage: A.has_rep(2)
False

The optional argument restrict may be used to restrict the possible images of the generators. To do so, restrict must be a tuple with entries of None, 'diagonal', 'lower', or 'upper'. Its length must match the number of generators of self.

sage: R.has_rep(2, restrict=[None, 'lower'])
True
sage: R.has_rep(2, restrict=['diagonal', 'upper'])
True

Use force=True if the function does not recognize the base field as computable, but the field is computable.

• has_irred_rep(n, gen_set=None, restrict=None, force=False)

Returns True if there exists an n-dimensional irreducible representation of self, and False otherwise. Of course, this function runs has_rep(n, restrict) to verify there is a representation in the first place, and returns False if not.

sage: R.<x,y> = FinitelyPresentedAlgebra(QQ, 'x*y + y*x')
sage: R.has_irred_rep(2)
True
sage: R.has_irred_rep(3)
False

The argument restrict may be used equivalenty to its use in has_rep().

sage: R.has_irred_rep(2, restrict=[None, 'lower'])
True
sage: R.has_irred_rep(2, restrict=['diagonal', 'upper'])
False

The argument gen_set may be set to 'PBW' or 'pbw', if self has an algebra basis similar to that of a Poincaré-Birkhoff-Witt basis.

sage: A.<a,b> = FinitelyPresentedAlgebra(QQ, 'b*a - a*b - 1')
sage: A.has_irred_rep(2, gen_set='pbw')
False

Alternatively, an explicit generating set for the algorithm implemented by this function can be given, as a tuple or array of FreeAlgebraElements. This is only useful if the package cannot reduce the elements of self, but they can be reduced in theory.

sage: A.<a,b> = FinitelyPresentedAlgebra(QQ, 'b*a - a*b - 1')
sage: A.has_irred_rep(2, gen_set = [A(a), A(b), A(a*b)])
False

Use force=True if the function does not recognize the base field as computable, but the field is computable.

• is_rep(image, n, force=False)

Returns True if the map generated by mapping the generators to the matrices defined in image is an n-dimensional representation of self, and False otherwise. The entries of image must be n-by-n matrices with entries in the algebraic closure of the base field of self. Its length must match the number of generators of self.

sage: R.<x,y> = FinitelyPresentedAlgebra(QQ, 'x*y + y*x')
sage: M1 = [[0, 1], [1, 0]]; M2 = [[0, -1], [1, 0]]
sage: R.is_rep([M1, M2], 2)
True

Use force=True if the function does not recognize the base field as computable, but the field is computable.

• is_irred_rep(image, n, force=False)

Returns True if the map generated by mapping the generators to the matrices defined in image is an n-dimensional irreducible representation of self, and False otherwise. Like above, the entries of image must be n-by-n matrices with entries in the algebraic closure of the base field of self. Its length must match the number of generators of self.

sage: R.<x,y> = FinitelyPresentedAlgebra(QQ, 'x*y + y*x')
sage: M1 = [[0, 1], [1, 0]]; M2 = [[0, -1], [1, 0]]
sage: R.is_irred_rep([M1, M2], 2)
True

Use force=True if the function does not recognize the base field as computable, but the field is computable.

If you use finitely_presented_algebra in your research, please cite this repository.

K. Rhoads. rhoadskj/finitely-presented-algebra: finitely-presented-algebra v1.0, 2019. https://github.com/rhoadskj/finitely-presented-algebra.

@software{fpa19,
   AUTHOR = {Rhoads, Kyle\v{s}},
    TITLE = {\texttt{rhoadskj/finitely-presented-algebra}: \texttt{finitely-presented-algebra} v1.0},
     NOTE = {\url{https://github.com/rhoadskj/finitely-presented-algebra}},
     YEAR = {2019},
}