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Primitive localizations of an enveloping algebra. (English) Zbl 0701.17005
Let L be a nonzero finite-dimensional Lie algebra over a field of characteristic zero, and U(L) its universal enveloping algebra with skew field of fractions D(L). The subspace of U(L), D(L) spanned by eigenvectors with respect to the action of ad L is called the semicenter, written Sz(U(L)), Sz(D(L)) respectively. These are graded algebras, the homogeneous elements being the semiinvariants, with grading semigroup \(\Lambda\) (L) consisting of the weights. Let \(\Lambda^ u(L)\) be the subgroup of units, i.e. weights \(\lambda\) such that -\(\lambda\) is also a weight, and \(Sz^ u(U(L))\) the corresponding subalgebra. The authors show that \(Sz^ u(U(L))\) is a unique factorization domain (UFD) but give an example to show that the center Z(U(L)) need not be a UFD. Further they prove the following conditions to be equivalent: (a) every non-zero ideal of U(L) meets the center of U(L) non-trivially, (b) \(\Lambda^ u(L)=\Lambda (L)\), (c) the localization of U(L) at its center is a simple ring.
The rest of the paper is a study of Z(U(L)) and Sz(U(L)), and of conditions for the localization of U(L) at its center to be primitive. Thus if T is a saturated multiplicaive set of invariants generating the center of D(L) (as field), then the subalgebra A of Sz(U(L)) generated by T is a UFD and Sz(U(L)) can be expressed as polynomial ring in a finite number of semiinvariants over A. The central localization of U(L) (for a non-abelian L) is primitive precisely when the center of D(L) is the field of fractions of the center of U(L); the authors also obtain conditions for the localization of U(L) at U(Z(L)) (the universal enveloping algebra of the center of L) to be primitive.
Reviewer: P.M.Cohn

17B35 Universal enveloping (super)algebras
16U30 Divisibility, noncommutative UFDs
16S30 Universal enveloping algebras of Lie algebras
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16U20 Ore rings, multiplicative sets, Ore localization
Full Text: DOI
[1] Anderson, D.F, Graded Krull domains, Comm. algebra, 7, 79-106, (1979) · Zbl 0402.13013
[2] Anderson, D.D; Anderson, D.F, Divisibility properties of graded domains, Canad. J. math., 34, 196-215, (1982) · Zbl 0437.13001
[3] Chevalley, C, ()
[4] Delvaux, L; Nauwelaerts, E; Ooms, A.I, On the semi-center of a universal enveloping algebra, J. algebra, 94, 324-346, (1985) · Zbl 0594.17010
[5] Dixmier, J, Enveloping algebras, () · Zbl 0867.17001
[6] Dixmier, J; Duflo, M; Vergne, M, Sur la représentation coadjointe d’une algébre de Lie, Compositio math., 25, No. 194, 309-323, (1974) · Zbl 0296.17009
[7] Elashvili, A.G, Frobenius Lie algebras, II, Trudy tbiliss. mat. inst. razmadze akad. nauk. gruzin. SSR, 77, 127-137, (1985) · Zbl 0626.17007
[8] Irving, R.S; Small, L.W, On the characterization of primitive ideals in enveloping algebras, Math. Z., 173, 217-221, (1980) · Zbl 0437.17002
[9] Jacobson, N, Basic algebra I, (1974), Freeman San Francisco · Zbl 0284.16001
[10] Krause, G.R; Lenagan, T.H, Growth of algebras and Gelfand-Kirillov dimension, () · Zbl 0564.16001
[11] Le Bruyn, L; Ooms, A.I, The semicenter of an enveloping algebra is factorial, (), 397-400 · Zbl 0558.17010
[12] Malliavin, M.P, Ultra produit d’algébres de Lie, (), 157-166 · Zbl 0512.17002
[13] Moeglin, C, Factorialité dans LES algébres enveloppantes, C. R. acad. sci. Paris A, 282, 1269-1272, (1976) · Zbl 0338.17002
[14] Moeglin, C, Idéaux bilatères dans LES algèbres enveloppantes, Bull. soc. math. France, 108, 143-186, (1980) · Zbl 0447.17008
[15] Montgomery, S, X-inner automorphisms of filtered algebras, (), 263-268 · Zbl 0474.16003
[16] Moore, C.C; Wolf, J.A, Square integrable representations of nilpotent groups, Trans. amer. math. soc., 185, 445-462, (1973) · Zbl 0274.22016
[17] Nauwelaerts, E; Ooms, A.I, Weights of semi-invariants of the quotient division ring of an enveloping algebra, (), 13-19 · Zbl 0663.17003
[18] Ooms, A.I, On Lie algebras having a primitive universal enveloping algebra, J. algebra, 32, 488-500, (1974) · Zbl 0355.17014
[19] Ooms, A.I, On Frobenius Lie algebras, Comm. algebra, 8, 13-52, (1980) · Zbl 0421.17004
[20] Rentschler, R; Vergne, M, Sur le semi-centre du corps enveloppant d’une algèbre de Lie, Ann. sci. école norm. sup (4), 6, 389-405, (1973) · Zbl 0293.17007
[21] Rentschler, R, Primitive ideals in enveloping algebras (general case), Math. surveys monographs, 24, (1987) · Zbl 0651.17005
[22] “Noetherian Rings and their Applications” (L. W. Small, Ed.), pp. 37-57, Amer. Math. Soc., Providence, R.I.
[23] Wauters, P, Factorial domains and graded rings, Comm. algebra, 17, 827-836, (1989) · Zbl 0675.13011
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