Polynomial identities for the Jordan algebra of \(2 \times 2\) upper triangular matrices
type: Item
instance of: scholarly article
title: Polynomial identities for the Jordan algebra of \(2 \times 2\) upper triangular matrices
zbMATH Open document ID: 1485.17032
author: Dimas José Gonçalves
author: Mateus Eduardo Salomão
author: Plamen Koshlukov
published in: Journal of Algebra
publication date: 2022-01-03T00:00:00Z
review text: Let \(K\) be an arbitrary field of any characteristic different from 2 and let \(UT_2(K)\) be the algebra of \(2\times 2\) upper triangular matrices over \(K\). Equipping \(UT_2(K)\) with the new product \(a\circ b=(1/2)(ab+ba)\) it becomes a Jordan algebra denoted by \(UJ_2(K)\). In the paper under review the authors study the Jordan polynomial identities of \(UJ_2(K)\). They find a basis of the polynomial identities. When the field \(K\) is infinite, the basis consists of three identities: \[ (x_1\circ x_2,x_3,x_4)-x_1\circ(x_2,x_3,x_4)-x_2\circ(x_1,x_3,x_4), \] \[ (x_1,(x_2,x_3,x_4),x_5),\quad (x_1,x_2,x_3)\circ(x_4,x_5,x_6), \] where \((u_1,u_2,u_3)\) is the associator. When \(\vert K\vert<\infty\), the authors add five more identities depending on the number of elements of \(K\). The authors also give an explicit basis as a vector space of the relatively free algebra of the variety \(\text{var}(UJ_2(K))\) generated by \(UJ_2(K)\). They also show that over an infinite field \(K\) the variety \(\text{var}(UJ_2(K))\) satisfies the Specht property, i.e. every subvariety of \(\text{var}(UJ_2(K))\) has a finite basis of its polynomial identities. The proof uses the method of Higman-Cohen based on the Higman lemma on partially ordered sets [\textit{D. E. Cohen}, J. Algebra 5, 267--273 (1967; Zbl 0157.34802)] for the proof of the Specht property for metabelian varieties of groups. The Specht property of finite Jordan algebras was established in [\textit{Yu. A. Medvedev}, Algebra Logic 18, 460--478 (1980; Zbl 0447.17012); translation from Algebra Logika 18, 723--748 (1979)]. In the 1960s and 1970s this was one of the few methods to prove the Specht property for varieties of groups, Lie and associative algebras. The introduction of the paper contains also an interesting survey on bases of polynomial identities and the Specht problem for associative, Lie and Jordan algebras.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 17C05
Mathematics Subject Classification ID: 16R10
zbMATH DE Number: 7451319
zbMATH Keywords: Jordan algebra
zbMATH Keywords: polynomial identities
zbMATH Keywords: upper triangular matrices
zbMATH Keywords: Specht property
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2021.11.020
OpenAlex ID: W3215671579
cites work: Representability and Specht problem for \(G\)-graded algebras.
cites work: Q5151810
cites work: Q3996716
cites work: Counterexamples to the Specht problem
cites work: A note on cocharacter sequence of Jordan upper triangular matrix algebra
cites work: Specht property for some varieties of Jordan algebras of almost polynomial growth
cites work: On identities in Lie algebras
cites work: A minimal basis of identities for a second-order matrix algebra over a field of characteristic 0
cites work: Q4700803
cites work: Q4761662
cites work: Ordering by Divisibility in Abstract Algebras
cites work: The Specht property of the ideals of identities of certain simple nonassociative algebras
cites work: Q4006519
cites work: Q3129435
cites work: Q5622399
cites work: VARIETIES AND $ Z_2$-GRADED ALGEBRAS
cites work: Finite basis property of identities of associative algebras
cites work: Basis of the identities of the matrix algebra of order two over a field of characteristic \(p\neq 2\)
cites work: Polynomial identities for the Jordan algebra of upper triangular matrices of order 2
cites work: The Polynomial Identities of the Grassman Algebra
cites work: Q5596920
cites work: Basis for identities of the algebra of upper triangular matrices
cites work: A taste of Jordan algebras
cites work: Identities of the tensor square of a Grassmann algebra
cites work: Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero
cites work: CONSTRUCTION OF NON-FINITELY BASED T-IDEALS
cites work: Gesetze in Ringen. I
cites work: Identities of PI-Algebras Graded by a Finite Abelian Group
cites work: Finitely generated algebras with involution and their identities.
cites work: Identities with involution for \(2\times 2\) upper triangular matrices algebra over a finite field
cites work: Q3470628
cites work: A finite basis for polynomial identities of the Jordan algebra of a bilinear form
cites work: VARIETIES OF LIE ALGEBRAS
cites work: Abelian-By-Nilpotent Varities of Lie Algebras
DOI: 10.1016/J.JALGEBRA.2021.11.020
Recommended article: Specht property for some varieties of Jordan algebras of almost polynomial growth
Recommended article: Polynomial identities for the Jordan algebra of upper triangular matrices of order 2
Recommended article: Q3202277
Recommended article: A note on cocharacter sequence of Jordan upper triangular matrix algebra
Recommended article: Variety of Jordan algebras \(\text{var}\,(UT_2(F)^{(+)})\) has almost polynomial growth
Recommended article: Q4133772
Recommended article: The algebra of \(2 \times 2\) upper triangular matrices as a commutative algebra: gradings, graded polynomial identities and Specht property
Recommended article: Specht property for the 2-graded identities of the Jordan algebra of a bilinear form
Recommended article: The Specht property of the ideals of identities of certain simple nonassociative algebras
Recommended article: Weak identities in the algebra of symmetric matrices of order two
description: scientific article; zbMATH DE number 7451319