Bell's primeness criterion and the simple Lie superalgebras (Q1295635)

Let \(L=L_0+L_1\) be a finite dimensional simple Lie superalgebra over an algebraically closed field \(k\) of characteristic zero. The following property was considered by \textit{A. D. Bell} [J. Pure Appl. Algebra 69, 111-120 (1990; Zbl 0723.17011)]. If \(f_1,\ldots,f_s\) is a base of \(L_1\), then the square matrix \(([f_i,f_j])\) of size \(s\) is non-singular over the symmetric algebra \(S(L_0)\). It was shown in the mentioned paper that under this assumption the universal enveloping algebra \(U(L)\) is prime. The main result of the present paper states that the algebra \(L\) satisfies this condition if and only if \(L\) is not one of: \(b(n),n\geq 3\); \(W(n)\) for odd \(n\geq 5\); \(S(n)\) for odd \(n\geq 3\). Of the exception above, \(U(b(n))\) and \(U(S(n))\) are not semiprime.