Codimensions of identities of solvable Lie superalgebras (Q6639661)

Let \(L\) be a finite dimensional Lie superalgebra over a field of characteristic 0. By analogy with the case of associative PI-algebras one defines the graded codimensions \(c_n^{\text{gr}}(L)\), \(n\geq 1\), and the graded PI-exponent\N\[\N\exp^{\text{gr}}(L)=\lim_{n\to\infty}\sqrt[n]{c_n^{\text{gr}}(L)}.\N\]\NIn the case of PI-algebras the codimensions and the PI-exponent behave in a nice way, but the situation is completely different for Lie superalgebras. For example the graded PI-exponent may exist and may be not an integer. In the special case when \(L\) is solvable and the commutator ideal \(L'\) is nilpotent, then the graded PI-exponent is an integer. \N\NIn their paper [J. Lie Theory 28, No. 4, 1189--1199 (2018; Zbl 1441.17006)] the authors of the paper under review constructed a series of solvable Lie algebras \(S(t)\), \(t\geq 2\), consisting of \(2\times 2\) block matrices with entries of \(t\times t\) upper triangular matrices subject to some restrictions in terms of an involution of the \(t\times t\) matrices. The constructed algebras have the property that the commutator ideal is not nilpotent. In the above cited paper the authors computed the graded PI-exponent of the Lie superalgebra \(S(2)\). The main result of the present paper gives the graded PI-exponent for all the series:\N\[\N\exp^{\text{gr}}(S(t))=\begin{cases}2t,\text{ if }t\text{ is even,}\\\N2t-1,\text{ if }t\text{ is odd.}\end{cases}.\N\]