Jordan algebras of a degenerate bilinear form: Specht property and their identities

Let K be a field and let Jn,k be the Jordan algebra of a degenerate symmetric bilinear form b of rank n - k over K. Then one can consider the decomposition Jn,k = Bn-k (R) Dk, where Bn-k represents the corresponding Jordan algebra, denoted as Bn-k = K (R) V. In this algebra, the restriction of b on the (n - k)-dimensional subspace V is non-degenerate, while Dk accounts for the degenerate part of Jn,k. This paper aims to provide necessary and sufficient conditions to check if a given multilinear polynomial is an identity for Jn,k. As a consequence of this result and under certain hypothesis on the base field, we exhibit a finite basis for the T-ideal of polynomial identities of Jn,k. Over a field of characteristic zero, we also prove that the ideal of identities of Jn,k satisfies the Specht property. Moreover, similar results are obtained for weak identities, trace identities and graded identities with a suitable Z2-grading as well. In all of these cases, we employ methods and results from Invariant Theory. Finally, as a consequence from the trace case, we provide a counterexample to the embedding problem given in [8] in case of infinite dimensional Jordan algebras with trace.