Journal of Pure and Applied Algebra 38 (1985) 159-166 North-Holland 159 zyxwvutsr ZERO DIVISORS IN ENVELOPING ALGEBRAS OF GRADED LIE ALGEBRAS Marc AUBRY and Jean-Michel LEMAIRE Laboratoire de MathPmatiques, (/.A. CNRS 168, UniversitP de Nice, Pare Valrose, F-06034 Nice CPdex, France Communicated by C. Lijfwall Received 19 January 1985 Dedicated to Jan-Erik Roos on his 50.th birthday Introduction Let L be a graded Lie algebra over a field k of characteristic different from 2. If L is concentrated in even degrees, that is, if L is a Lie algebra in the ordinary sense, its enveloping algebra UL has no zero divisors: this is an easy consequence of the PoincarC-Birkhoff-Witt theorem. On the other hand, if L contains a non- zero element of odd degree x such that [x, x] = 0, clearly UL has zero divisors. Following R. Bprgvad [3], let us say that a graded Lie algebra is ‘torsion-free’ if [x, x] #0 for every non-zero x of odd degree, and ‘absolutely torsion-free’ if it re- mains torsion-free after field extension to the algebraic closure L of k. Then our main result states: Theorem. The enveloping algebra UL of an absolutely torsion free graded Lie algebra L has no zero divisors. We also show examples of torsion-free Lie algebras over the reals whose envelop- ing algebras have zero-divisors. The proof takes two steps: we first reduce to the case when L is concentrated in degrees 1 and 2 with dim L, = dim L, = n < 00 (the ‘(n, n)-quadratic’ case). Next we prove the theorem by induction on II, using suitable (non-commutative) localization and the fact, due to BDgvad, that a solvable graded Lie algebra has finite global dimension if and only if it is absolutely torsion free. Our interest in this question grew out from the analogy between regular sequences in commutative algebra and ‘inert’ sequences in Lie algebras developed in [l] and [5]. As a matter of fact, the reduction to the quadratic case (Proposition 2.1) was obtained by S. Halperin and the second author during the preparation of [5]. We also wish to acknowledge the kind cooperation of several geometers in Nice, 0022-4049/85/$3.30 0 1985, Elsevier Science Publishers B.V. (North-Holland)