Polynomial identities for the Jordan algebra of 2 × 2 upper triangular matrices

Abstract

Let K be a field (finite or infinite) of char(K)≠2 and let UT2(K) be the 2×2 upper triangular matrix algebra over K. If ⋅ is the usual product on UT2(K) then with the new product a∘b=(1/2)(a⋅b+b⋅a) we have that UT2(K) is a Jordan algebra, denoted by UJ2=UJ2(K). In this paper, we describe the set I of all polynomial identities of UJ2 and a linear basis for the corresponding relatively free algebra. Moreover, if K is infinite we prove that I has the Specht property. In other words I, and every T-ideal containing I, is finitely generated as a T-ideal.