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Factorial properties of the enveloping algebra of a nilpotent Lie algebra in prime characteristic

2006; Elsevier BV; Volume: 308; Issue: 1 Linguagem: Inglês

10.1016/j.jalgebra.2006.08.034

1090-266X

Amiram Braun,

Advanced Algebra and Geometry

Let U ( L ) be the enveloping algebra of a finite-dimensional nilpotent Lie algebra L , over a prime characteristic field. We prove that its center Z ( U ( L ) ) is a unique factorization (UFD). We also show that U ( L ) has a non-commutative UFD property, namely, each height one prime ideal in U ( L ) is generated by a central element. We prove both results simultaneously, using non-commutative (PI, maximal order) technique. Our results are prime characteristic analogues of similar ones in characteristic zero, which are due to Dixmier [J. Dixmier, Sur l'algèbre enveloppante d'une algebra de Lie nilpotente, Arch. Math. 10 (1959) 321–32] and Moeglin [C. Moeglin, Factorialité dans les algèbres enveloppantes, C. R. Acad. Sci. Paris (A) 282 (1976) 1269–1272]. We have recently applied these results to show that U ( L ) is a Calabi–Yau algebra.