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Zero divisors in enveloping algebras of graded Lie algebras. (English) Zbl 0587.55006

Let k be a field of characteristic different from 2. Following R. Bøgvad [Astérisque 113/114, 156-166 (1984; Zbl 0552.17012)], a graded Lie algebra over k, \(L=\oplus_{i\geq 1}L_ i\), is torsion free if [x,x]\(\neq 0\) for every nonzero x of odd degree and absolutely torsion free if it remains torsion free after field extension to the algebraic closure \(\bar k\) of k. The authors prove: the enveloping algebra UL of an absolutely torsion free graded Lie algebra L has no zero divisors.
In an appendix they reproduce a different proof due to R. Bøgvad.
Reviewer: J.C.Thomas

MSC:

55Q52 Homotopy groups of special spaces
57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
17B70 Graded Lie (super)algebras
17B35 Universal enveloping (super)algebras

Citations:

Zbl 0552.17012
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References:

[1] Anick, D., Non-commutative algebras and their Hilbert series, J. Algebra, 78, 120-140 (1982) · Zbl 0502.16002
[2] Borho, W.; Rentschler, R., Oresche Teilmengen in Einhüllenden Algebren, Math. Ann., 217, 201-210 (1975) · Zbl 0297.17004
[3] R. Bøgvad, Some elementary results on the cohomology of graded Lie algebras, in: Homotopie Algébrique et Algèbre Locale, Astérisque 113-114, 156-166.; R. Bøgvad, Some elementary results on the cohomology of graded Lie algebras, in: Homotopie Algébrique et Algèbre Locale, Astérisque 113-114, 156-166. · Zbl 0552.17012
[4] Halperin, S., Finiteness in the minimal models of Sullivan, Trans. A.M.S., 230, 173-199 (1977) · Zbl 0364.55014
[5] Halperin, S.; Lemaire, J. M., Suites inertes dans les algèbres de Lie graduées, (Prépublications mathématiques (1984), Université de Nice), no. 22 · Zbl 0655.55004
[6] Milnor, J.; Moore, J. C., On the structure of Hopf algebras, Ann. Math., 81, 211-264 (1965) · Zbl 0163.28202
[7] Stenström, B., Rings of Quotients, (Grundl. d.Math. Wiss, 217 (1975), Springer: Springer Berlin) · Zbl 0194.06602
[8] Sjödin, G., A set of generators for \(Ext_R (k, k)\), Math. Scand., 38, 199-210 (1976) · Zbl 0346.18017
[9] Wall, C. T.C., Nets of conics, Math. Proc. Cambridge Ph. Soc., 81, 351-364 (1977) · Zbl 0351.14032
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