Centers of higher degree trace forms

Abstract

If θ is a regular, symmetric d-linear form on a vector space V, the center of (V,θ) is the set of linear maps f:V→V symmetric relative to θ. If d>2, it is well known that this center is a commutative subalgebra of End(V).

When A is a Frobenius algebra with “trace” ℓ, we investigate the trace form φ(x)=ℓ(xd) on A. When A is commutative, A itself is the center of that trace form and the orthogonal group O(V,φ) is closely related to the automorphism group of the algebra A. In non-commutative cases, trace forms are more difficult to analyze. If A is a symmetric algebra, the center of the degree d trace form on A turns out to be N(A+), the nucleus of the induced Jordan algebra