Bilinear Forms on Frobenius Algebras

Abstract

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