On identities of infinite dimensional Lie superalgebras (Q2855892)

Let \(F\) be an algebraically closed field of characteristic 0, the authors study the numerical invariants of polynomial identities for Lie superalgebras. One of the most important invariants is the codimension sequence \(c_n(A)\) of the algebra \(A\). It is well known that if \(A\) is an associative algebra then \(c_n(A)\) is exponentially bounded. For Lie algebras the situation is more complicated: there exist Lie algebras \(L\) satisfying identities and such that \(c_n(L)\) is not exponentially bounded. On the other hand, for large and important classes of Lie algebras it is known that the codimension growth is exponential. These include finite dimensional algebras, Virasoro algebras, simple Lie algebras of Cartan type, Kac-Moody algebras.NEWLINENEWLINELet \(c_n(A)\) be exponentially bounded, then one may consider \(\liminf (c_n(A))^{1/n}\) and \(\limsup (c_n(A))^{1/n}\); if these coincide then the limit of the corresponding sequence is called the PI exponent of \(A\), denoted by \(\exp(A)\). Amitsur's conjecture states that in the case of associative algebras \(\exp(A)\) exists and is an integer; this was shown to be true by Giambruno and second-named author of the present paper. Although there is an analogous result for large classes of Lie algebras there are counter-examples in the case of Lie algebras. If \(L\) is a Lie superalgebra then \(\exp(L)\) may exist and be a non-integer.NEWLINENEWLINEThe main result of the paper under review is the following theorem. Assume \(L=L_0\oplus L_1\) is a 2-graded simple finite dimensional Lie algebra and let \(\widetilde{L} = L_0\otimes G_0\oplus L_1\otimes G_1\) be its Grassmann envelope, then \(\exp(\widetilde{L})\) exists and is an integer. More precisely, \(\exp(\widetilde{L}) = \dim L\).NEWLINENEWLINEAdditionally the authors establish a similar result for the 2-graded codimension sequence.