| A finite-dimensional algebra R over a field k is Frobenius (respectively, symmetric) if R ≃ Rˆ := HoMk (R, k) as right R-modules ((R,R)-bimodules). Equivalently, R possesses a nondegenerate associative (symmetric) k-bilinear form B : R × R → k. Discovered a century ago via group representations, Frobenius algebras have been the subject of renewed study because of recently discovered applications to quantum physics and coding theory.; Because there is a known ring-theoretic description of (Rˆ )R, the property of R being Frobenius is independent of the field k. We ask if the symmetric property is too and if the k-dual of an arbitrary right R-module is independent of k. These and several similar questions are interdependent, and the key to answering them is the Nakayama automorphism defined by B. We prove that the Nakayama automorphism is independent of k, answering all the questions for Frobenius algebras. We define a ring-theoretic generalization of symmetric k-algebras.; Next we ask to what extent the homothety type of the form B is unique. If R is local symmetric with residue field k, then R has a unique symmetric form, and it has a unique form if it is commutative. If R is Frobenius and the Nakayama automorphism has inner order n ≠ 0 in k, we can construct an automorphism with finite order. This defines a norm on R that gives a necessary condition for two forms to be homothetic. The converse is a conjecture, which we prove for algebras whose maximal ideals have low index of nilpotence, in which case the result implies that all forms are homothetic. Finally, we show that if dimk R is even, then det B is an invariant for R.; Lastly, we classify low-dimensional local commutative algebras in terms of extension fields of k, the square class group, and quadratic forms. An invariant on four-dimensional algebras indicates which are Frobenius. Finally, we show that a five-dimensional Frobenius algebra with residue field k is determined completely by a Pfister form and the index of nilpotence of its maximal ideal. |
