Group Algebras

This module implements the category of group algebras for arbitrary groups over arbitrary commutative rings. For details, see sage.categories.algebra_functor.

AUTHOR:

  • David Loeffler (2008-08-24): initial version
  • Martin Raum (2009-08): update to use new coercion model – see trac ticket #6670.
  • John Palmieri (2011-07): more updates to coercion, categories, etc., group algebras constructed using CombinatorialFreeModule – see trac ticket #6670.
  • Nicolas M. Thiéry (2010-2017), Travis Scrimshaw (2017): generalization to a covariant functorial construction for monoid algebras, and beyond – see e.g. trac ticket #18700.
sage.categories.group_algebras.GroupAlgebras

The category of group algebras over a given base ring.

EXAMPLES:

sage: C = Groups().Algebras(ZZ); C
Category of group algebras over Integer Ring
sage: C.super_categories()
[Category of hopf algebras with basis over Integer Ring,
 Category of monoid algebras over Integer Ring]

We can also construct this category with:

sage: C is GroupAlgebras(ZZ)
True

Here is how to create the group algebra of a group \(G\):

sage: G = DihedralGroup(5)
sage: QG = G.algebra(QQ); QG
Algebra of Dihedral group of order 10 as a permutation group over Rational Field

and an example of computation:

sage: g = G.an_element(); g
(1,4)(2,3)
sage: (QG.term(g) + 1)**3
4*() + 4*(1,4)(2,3)

Todo

  • Check which methods would be better located in Monoid.Algebras or Groups.Finite.Algebras.