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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

MSC:
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
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