Self-dual and quasi self-dual algebras

A self-dual algebra is an associative or Lie algebra A together with an A bimodule isomorphism A → A∨op, where A∨ = Homk(A, k), the dual bimodule to A (considered as an A bimodule), and A∨op is the the same underlying k module as A∨ but is an A bimodule whose left operation by an element a ∈ A is the same as the right operation by a on A∨, and similarly with left and right interchanged. This induces an isomorphism H∗(A,A) ∼= H∗(A,A∨ ); algebras with such an isomorphism are quasi self-dual. For these algebras H∗(A,A) is a contravariant functor of A. They form a full subcategory of the category of the category of associative or Lie algebras, respectively. Finite dimensional associative self-dual algebras over a field are identical with symmetric Frobenius algebras (which are closely connected to 1+1 dimensional topological quantum field theory). Finite poset algebras are quasi self-dual. For an important class of algebras A the cohomology H∗(A,A) of A with coefficients in itself is, remarkably, a contravariant functor of A. Many of these algebras are related to geometric objects. It is a familiar fact that for most categories of geometric objects cohomology is a contravariant functor. The cohomology of algebras behaves somewhat perversely. If A is an associative algebra over a (commutative, associative, unital) ring k and M an A-bimodule then the Hochschild cohomology H∗(A,M) is a contravariant functor of A but a covariant functor of M , and similarly with the Eilenberg-Chevalley cohomology of Lie algebras (but note that the first instance of Lie cohomology with coefficients in a non-trivial module seems to be due, again, to Hochschild). One of the most important A-bimodules is A itself, for H∗(A,A) has a rich structure and governs in particular the deformations of A, but in general H∗(A,A) is neither a covariant nor contravariant functor of A. For an important class of algebras, however, H∗(A,A) is indeed a contravariant functor of A. These are the quasi self-dual ones, those for which there is an isomorphism H∗(A,A) ∼= H∗(A,A∨ ). We consider here only algebras A which are finite free modules over k, and likewise for A bimodules, but some of the most interesting cases are likely to be infinite dimensional, requiring topological considerations. For categories of algebras where it is meaningful to consider the cohomology H∗(A, k) of A with coefficients in k as a trivial module this cohomology, which