Polynomial identities for the Jordan algebra of \(2 \times 2\) upper triangular matrices
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.
scientific article; zbMATH DE number 7451319
scientific article