Recall that for k a field, a finite dimensional k-algebra A is called *symmetric* if it is isomorphic to its dual as a bimodule of itself. Which is to say, there's a trace map t:A -> k such that t(ab)=t(ba) and for any nonzero a, there is a b such that t(ab) is not 0.

The most popular examples of symmetric algebras are the cohomology rings of compact manifolds. In this case, the trace is integration of top forms.

Various algebras arising in representation theory are symmetric and this has some nice consequences for their categories of representations (for example, the Serre functor is trivial).

Khovanov and Lauda have defined some algebras by explicit generators and relations (see, for example 0909.1810, the cyclotomic KLR algebras, which I strongly believe are symmetric, but after quite a bit of trying, have had no luck with. Anyone else care to try their hand?