Self-dual and quasi self-dual algebras

TL;DR

This paper explores the properties of self-dual and quasi self-dual algebras, linking their cohomology to deformation theory and classifying finite-dimensional cases, including poset algebras.

Contribution

It characterizes self-dual and quasi self-dual algebras, relates them to Frobenius algebras, and shows all finite poset algebras are quasi self-dual.

Findings

Finite dimensional self-dual algebras are symmetric Frobenius algebras.

Cohomology with coefficients in the algebra is a contravariant functor.

All finite poset algebras are quasi self-dual.

Abstract

A self-dual algebras is one isomorphic as a module to the opposite of its dual; a quasi self-dual algebra is one whose cohomology with coefficients in itself is isomorphic to that with coefficients in the opposite of its dual. For these algebras, cohomology with coefficients in itself, which governs its deformation theory, is a contravariant functor of the algebra. Finite dimensional self-dual algebras over a field are identical with symmetric Frobenius algebras. (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.