## 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

### Keywords:

Cartan Lie superalgebra; universal enveloping algebra

Zbl 0723.17011
Full Text:

### References:

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