Self-dual and quasi self-dual algebras

Abstract

Self-dual algebras are ones with an A bimodule isomorphism A → A ∨op, where A ∨ = Hom k (A, k) and A ∨op is the same underlying k-module as A ∨ but with left and right operations by A interchanged. These are in particular quasi self-dual algebras, i.e., ones with an isomorphism H*(A,A) ≌ H*(A,A ∨ op). For all such algebras H*(A,A) is a contravariant functor of A. Finite-dimensional associative self-dual algebras over a field are identical with symmetric Frobenius algebras; an example of deformation of one is given. (The monoidal category of commutative Frobenius algebras is known to be equivalent to that of 1+1 dimensional topological quantum field theories.) All finite poset algebras are quasi self-dual.