On a variety of right-symmetric algebras (Q6597519)

In 1950, Specht asked whether any variety of associative algebras has a finite basis of identities (or, equivalently, whether the ideal of identities of any associative algebra is finitely generated as a \(T\)-ideal). This question became a driving force in the development of the theory of polynomial identities in algebras, culminating in Kemer's positive solution to the problem in the mid-1980s for algebras over fields of characteristic zero. It is worth noting, however, that the problem has a negative solution in the case of positive characteristic.\N\NSubsequently, the Specht problem was extended to other varieties of algebras. A variety of algebras \(\mathcal{V}\) satisfies the \emph{Specht property} if every subvariety of \(\mathcal{V}\) has a finite basis of identities. Many researchers have studied the Specht problem for different varieties, employing various techniques.\N\NIt is worth pointing out that the case of Lie algebras remains open.\N\NIn the paper under review, the authors investigate the Specht property for right-symmetric algebras over arbitrary fields. Specifically, they construct a finite-dimensional metabelian right-symmetric algebra whose ideal of identities is not finitely based as a \(T\)-ideal.\N\NRecall that a variety of algebras is called right-symmetric if it satisfies\N\[\N(x, y, z) = (x, z, y),\N\]\Nwhere \((a, b, c) = (ab)c - a(bc)\) denotes the associator of \(a\), \(b\), and \(c\).\N\NRight-symmetric algebras are significant because they are Lie-admissible, meaning that they form a Lie algebra with respect to the commutator.\N\NThe algebra constructed in the paper, denoted \(P_2\), is 10-dimensional and satisfies the following identities:\N\[\N[[x, y], z] = 0, \quad (xy)x = 0, \quad \text{and} \quad (xy)(zw) = 0.\N\]\NThe last identity is referred to as the metabelian identity.\N\NThe construction given in the paper is based on methods introduced by \textit{I. V. L'vov} [Sib. Mat. Zh. 19, 91--99 (1978; Zbl 0388.16014); translation in Sib. Math. J. 19, 63-69 (1978). ] and further developed by \textit{I. M. Isaev} [Algebra Logic 25, 86--96 (1986; Zbl 0618.17010); translation from Algebra Logika 25, No. 2, 136--153 (1986)].