Regular Representation - More General Algebras

More General Algebras

The regular representation of a group ring is such that the left-hand and right-hand regular representations give isomorphic modules (and we often need not distinguish the cases). Given an algebra over a field A, it doesn't immediately make sense to ask about the relation between A as left-module over itself, and as right-module. In the group case, the mapping on basis elements g of K defined by taking the inverse element gives an isomorphism of K to its opposite ring. For A general, such a structure is called a Frobenius algebra. As the name implies, these were introduced by Frobenius in the nineteenth century. They have been shown to be related to topological quantum field theory in 1 + 1 dimensions.