Frobenius Algebra - Examples

Examples

1. Any matrix algebra defined over a field k is a Frobenius algebra with Frobenius form σ(a,b)=tr(a·b) where tr denotes the trace.
2. Any finite-dimensional unital associative algebra A has a natural homomorphism to its own endomorphism ring End(A). A bilinear form can be defined on A in the sense of the previous example. If this bilinear form is nondegenerate, then it equips A with the structure of a Frobenius algebra.
3. Every group ring of a finite group over a field is a Frobenius algebra, with Frobenius form σ(a,b) the coefficient of the identity element in a·b. This is a special case of example 2.
4. For a field k, the four-dimensional k-algebra k/ (x2, y2) is a Frobenius algebra in the sense of the second example.
5. For a field k not of characteristic 2, the three-dimensional k-algebra k/ (x, y)2 is not a Frobenius algebra in the sense of the second example.