On codimension growth of finite-dimensional Lie superalgebras (Q2880401)

Let \(A\) be a not necessarily associative algebra over a field \(F\) of characteristic 0. Let \((c_n(A))_{n \geq 1}\) be the sequence of codimensions of \(A\), which is a measurement of the growth of polynomial identities on \(A\). If the sequence \(n \mapsto (c_n(A))^{1/n}\) converges, the value \(\exp(A) := \lim_{n \to \infty} (c_n(A))^{1/n}\) is called PI \textit{exponent of } \(A\). One classical question on this invariant is to find classes of algebras for which this value exists, and to determine when it is an integer. For example, a classical result states that this is the case for any associative PI algebra [\textit{A. Regev}, Isr. J. Math. 11, 131--152 (1972; Zbl 0249.16007)]. More recently, existence and integrality of \(\exp(A)\) was proved for finite-dimensional Lie algebras in [\textit{M. V. Zaitsev}, Izv. Math. 66, No. 3, 463--487 (2002; Zbl 1057.17003)]; however for infinite-dimensional Lie algebras, the PI exponent can be any real value \(\alpha > 1\) [\textit{A. Giambruno}, \textit{S. Mishchenko} and \textit{M. Zaicev}, Adv. Math. 217, No. 3, 1027--1052 (2008; Zbl 1133.17001)].NEWLINENEWLINENow assume that \(F\) is algebraically closed and \(A\) is finite-dimensional over \(F\). In that situation, it is known that if \(A\) is associative, Lie, Jordan or alternative, then \(\exp(A)\) is an integer, bounded above by \(\dim(A)\), with equality if and only if \(A\) is simple. In the paper under review, the authors prove that for arbitrary finite-dimensional simple algebras, the PI exponent \(\exp(A)\) exists and is bounded above by \(\dim(A)\). Furthermore, they exhibit a class of simple finite-dimensional Lie superalgebras \(\mathfrak{b}_n\) such that \(\exp(\mathfrak{b}_n) < \dim(\mathfrak{b}_n)\). Finally, for the 7-dimensional Lie superalgebra \(\mathfrak{b}_2\), they show that \(6 < \exp(\mathfrak{b}_2) < 7\).