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Bell’s primeness criterion for \(W(2n+1)\). (English) Zbl 0892.17012

The paper deals with a problem in Lie superalgebra theory. A result due to A. D. Bell [J. Pure Appl. Algebra 69, 111-120 (1990; Zbl 0723.17011)] relates the primeness of the universal enveloping algebra of a Lie superalgebra to the nonsingularity of the product matrix. Among the Cartan Lie superalgebras, the series \(W(n)\) for odd \(n\) remains a class for which this nonsingularity has not yet been investigated, and where other approaches have failed. In this paper, the authors develop experimental (or computational) methods and apply them to \(W(5)\), the case of \(W(3)\) being relatively easy. The interesting fact is that the experimental methods for \(W(5)\) not only lead to an unambiguous conclusion, but finally to a rigorous proof that the product matrix is singular for all \(W(n)\) with odd \(n\geq 5\).

MSC:

17B35 Universal enveloping (super)algebras
17A70 Superalgebras
17-08 Computational methods for problems pertaining to nonassociative rings and algebras

Citations:

Zbl 0723.17011
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References:

[1] DOI: 10.1016/0022-4049(90)90036-H · Zbl 0723.17011
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