Specht property for some varieties of Jordan algebras of almost polynomial growth (Q1755554)

Let \(F\) be a field of characteristic 0, and let \(A=UT_2(F)\) stand for the associative algebra of \(2\times 2\) upper triangular matrices. One obtains on \(A\) the structure of a Jordan algebra \(UJ_2\) by means of the Jordan product \(a\circ b=(ab+ba)/2\) for every \(a\), \(b\in A\). The Jordan algebra \(UJ_2\) admits several gradings by the cyclic group \(C_2\) of order 2. There are, up to isomorphism, three non-isomorphic non-trivial gradings. If one adds the trivial grading, and another grading by \(C_2\times C_2\), one gets the list of all gradings on \(UJ_2\). These gradings and the corresponding graded identities were described in [\textit{P. Koshlukov} and \textit{F. Martino}, J. Pure Appl. Algebra 216, No. 11, 2524--2532 (2012; Zbl 1287.17053)]. The main result of the paper under review is that in each of these five cases, the ideal of the (graded) identities of \(UJ_2\) satisfies the Specht property. This means that every ideal of (graded) identities containing the one of \(UJ_2\), is finitely generated as an ideal of (graded) identities. Additionally the authors prove that a metabelian Jordan algebra also satisfies the Specht property.