Specht property for the graded identities of the pair \((M_2 (D), sl_2 (D))\) (Q6592911)

Let \(D\) be a Noetherian infinite integral domain, let \(M_2(D)\) be the algebra of \(2\times 2\) matrices with entries in \(D\) and let \(sl_2(D)\) be the Lie algebra of the traceless matrices in \(M_2(D)\). Both \(M_2(D)\) and \(sl_2(D)\) are equipped with the natural \({\mathbb Z}_2\)-grading, \({\mathbb Z}_2=\{0,1\}\). A polynomial \(f(Y,Z)\) in the free graded associative \(D\)-algebra \(D\langle Y,Z\rangle\) (with \(0\)-graded variables \(Y\) and \(1\)-graded variables \(Z\)) is a weak polynomial identity for the pair \((M_2(D),sl_2(D))\) if it vanishes in \(M_2(D)\) when evaluated on \(sl_2(D)\). \par The first main result of the paper under review states that all graded weak polynomial identities of the pair \((M_2(D),sl_2(D))\) follow from its graded weak identities\N\[\N[y_1,y_2],\, z_1z_2z_3 - z_3z_2z_1,\, yz+zy.\N\]\NIn the proof the authors obtain also a basis as a free \(D\)-module of \(D\langle Y,Z\rangle\) modulo the graded weak identities of \((M_2(D),sl_2(D))\). \N\N\NThe next main result is that the graded weak identities of the pair \((M_2(D),sl_2(D))\) satisfy the Specht property, i.e., the graded weak identities of any pair satisfying the graded identities of \((M_2(D),sl_2(D))\) follow from a finite number of them. The proof is based on the technique of partially well ordered sets developed by \N\textit{G. Higman} in [Proc. Lond. Math. Soc. (3) 2, 326--336 (1952; Zbl 0047.03402)] and used intensively for the proof of the Specht property for different classes of groups and algebras. \par The introduction of the paper contains a very informative survey on ordinary and weak polynomial identities for associative and Lie algebras in the spirit of the obtained in the paper results.