Bilinear Forms on Frobenius Algebras

We analyze the homothety types of associative bilinear forms that can occur on a Hopf algebra or on a local Frobenius $k$-algebra $R$ with residue field $k$. If $R$ is symmetric, then there exists a unique form on $R$ up to homothety iff $R$ is commutative. If $R$ is Frobenius, then we introduce a norm based on the Nakayama automorphism of $R$. We show that if two forms on $R$ are homothetic, then the norm of the unit separating them is central, and we conjecture the converse. We show that if the dimension of $R$ is even, then the determinant of a form on $R$, taken in $\dot k/\dot k^2$, is an invariant for $R$. \textit{Key words}: bilinear form, Frobenius algebra, homothety, Hopf algebra, isometry, local algebra, Nakayama automorphism, Ore extension, symmetric algebra