Polynomial identities for the Jordan algebra of a symmetric bilinear form
Let K be a field of characteristic 0. Let \(V_ k\) be a vector space of dimension k over K with a nondegenerate symmetric bilinear form f. For \(k=\infty\) the subscript k is usually omitted. The form f determines a structure of a Jordan algebra on \(G_ k=K+V_ k\). It is known that \(G_ k\) is a simple Jordan algebra. The author obtains a description of the relatively free algebras \(FJ_ m({\mathcal M}_ k)\) with m generators for arbitrary m of the variety \({\mathcal M}_ k=Var G_ k\). The \(GL_ m\)- module structure of these algebras is established and their Hilbert (or Poincaré) series are computed. The varieties \({\mathcal M}_ k\) are described as subvarieties of \({\mathcal M}\) and \({\mathcal M}_{k+1}\). Formulae for codimensions of \({\mathcal M}\) and \({\mathcal M}_ k\) are found and their growth is estimated asymptotically.
scientific article; zbMATH DE number 4023527
scientific article