Bilinear Forms on Frobenius Algebras

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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 k/ k², is an invariant for R. Key words: bilinear form, Frobenius algebra, homothety, Hopf algebra, isometry, local algebra, Nakayama automorphism, Ore extension, symmetric algebra