Editing Regular representation (section)

==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''[''G''] defined by taking the [[inverse element]] gives an isomorphism of ''K''[''G''] to its ''opposite'' ring. For ''A'' general, such a structure is called a [[Frobenius algebra]]. As the name implies, these were introduced by [[Ferdinand Georg Frobenius|Frobenius]] in the nineteenth century. They have been shown to be related to [[topological quantum field theory]] in 1 + 1 dimensions by a particular instance of the [[cobordism hypothesis]].