Polynomial identities for the Jordan algebra of a symmetric bilinear form
type: Item
instance of: scholarly article
title: Polynomial identities for the Jordan algebra of a symmetric bilinear form
zbMATH Open document ID: 0629.17011
DOI: 10.1016/0021-8693(87)90122-0
author: Vesselin Drensky
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 17C05
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 08B20
zbMATH DE Number: 4023527
zbMATH Keywords: variety of Jordan algebras
zbMATH Keywords: Hilbert series
zbMATH Keywords: Poincaré series
zbMATH Keywords: relatively free algebras
zbMATH Keywords: generators
zbMATH Keywords: codimensions
zbMATH Keywords: growth
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90122-0
OpenAlex ID: W2003681275
cites work: Homogeneous polynomial identities
cites work: Codimensions of T-ideals and Hilbert series of relatively free algebras
cites work: Q4005997
cites work: Q3658162
cites work: Q3960974
cites work: T-ideals containing all matrix polynomial identities
cites work: Invariants and the ring of generic matrices
cites work: PI-algebras. An introduction
cites work: Q5622399
cites work: Q3952291
cites work: Q3947818
cites work: Algebras satisfying a Capelli identity
cites work: Asymptotic values for degrees associated with strips of Young diagrams
cites work: Q3220818
cites work: On Symmetrized Kronecker Powers and the Structure of the Free Lie Ring
cites work: VARIETIES OF LIE ALGEBRAS
cites work: Q4435456
cites work: Q3947798
description: scientific article; zbMATH DE number 4023527