Primeness criteria for universal enveloping algebras of Lie color algebras (Q5925845)

Let \({\mathcal L}={\mathcal L}_++{\mathcal L}_-\) be a Lie color algebra over a field of characteristic different from 2, where \(\mathcal L\) is graded by an abelian group \(G\), \(\dim{\mathcal L}_-<\infty\) and \(\{y_1,\ldots,y_n\}\) is a basis of the odd part \({\mathcal L}_-\) of \(\mathcal L\). The Bell criterion [\textit{A. D. Bell}, J. Pure Appl. Algebra 69, 111-120 (1990; Zbl 0723.17011)] for the primeness of the universal enveloping algebra \(U({\mathcal L})\) of a Lie superalgebra \(\mathcal L\) states that if the determinant \(\det{\mathcal L}\) of the product matrix \((\langle y_i,y_j\rangle)\) considered as a matrix over the symmetric algebra of \({\mathcal L}_+\) is different from 0, then \(U({\mathcal L})\) is prime. NEWLINENEWLINENEWLINEIn the paper under review the author shows that for any color superalgebra \(\mathcal L\) the determinantal condition \(\det{\mathcal L}\not =0\) of Bell implies only that \(U({\mathcal L})\) is semiprime and gives examples when the algebra \(U({\mathcal L})\) is not prime. For primeness one needs additional conditions and the main result of the paper is to give a criterion for primeness. In particular, \(U({\mathcal L})\) is prime if the grading group \(G\) is either finite with cyclic 2-torsion subgroup or finitely generated and such that for each elementary divisor \(2^l\) of \(G\) the base field does not contain a primitive \(2^l\)-th root of unity. NEWLINENEWLINENEWLINEThe proofs involve iterated quadratic extensions of graded algebras, graded valuations and graded Cliffors algebras.