Factorial properties of the enveloping algebra of a nilpotent Lie algebra in prime characteristic
Let \(U(L)\) be the universal enveloping algebra of the finite dimensional nilpotent Lie algebra \(L\) over a field of prime characteristic. The author establishes that the centre \(Z(U(L))\) of \(U(L)\) is a unique factorization domain. He also proves a noncommutative UFD-property of \(U(L)\) showing that every height one prime ideal of \(U(L)\) is generated by a central element. Recently the author and Hajarnavis have established that, for any finite dimensional Lie algebra \(L\), the divisor class group of \(Z(U(L))\) is bounded torsion (the bound is the PI-degree of \(U(L)\)). Hence, the first result of the present paper may be regarded as a part of a program which is aimed at improving the division class group result, for more specialized Lie algebras. The second result is a noncommutative extension of the Auslander-Buchsbaum UFD-theorem to noncommutative smooth PI-algebras, since \(U(L)\) is an important example of such an algebra. Both results have their characteristic zero analogues due to Dixmier and Moeglin, respectively [\textit{J. Dixmier}, Arch. Math. 10, 321--326 (1959; Zbl 0146.26102) and \textit{C. Moeglin}, C. R. Acad. Sci., Paris, Sér. A 282, 1269--1272 (1976; Zbl 0338.17002)]. Nevertheless the proofs are quite different from the case of characteristic zero because one cannot use the symmetrization map. Instead, the author uses noncommutative techniques, including algebras with polynomial identities and maximal orders.
scientific article; zbMATH DE number 5129702
scientific article