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On the semi-center of a universal enveloping algebra. (English) Zbl 0594.17010
Let $$U$$ denote the universal enveloping algebra of a non-zero finite dimensional Lie algebra $${\mathfrak g}$$ over a field $$k$$ of characteristic zero and let $$D$$ denote the division ring of quotients of $$U$$. Let $$k'$$ denote the algebraic closure of $$k$$ and $$U'$$ the universal enveloping algebra of $${\mathfrak g}'={\mathfrak g}\otimes_k k'.$$ Using the fact that the semi-center $$\mathrm{Sz}(U')$$ is a UFD [C. Moeglin, C. R. Acad. Sci., Paris, Sér. A 282, 1269–1272 (1976; Zbl 0338.17002)] it is proved that the ring $$U\cap \mathrm{Sz}(U')$$ is also a UFD and hence any non-zero semi-invariant of $$U$$ can be factored in a unique way as a product of irreducible semi-invariants.
Another result of C. Moeglin [Bull. Soc. Math. Fr. 108, 143–186 (1980; Zbl 0447.17008)] generalized independently by M. P. Malliavin [Lect. Notes Math. 924, 157–166 (1982; Zbl 0482.17003)] and V. A. Ginzburg [On the ideals of U($${\mathfrak g})$$ (to appear)] states that any non-zero ideal of $$U$$ contains a non-zero semi-invariant and this fact is also extremely useful.
Let $$\Lambda({\mathfrak g})$$ (respectively $$\Lambda_ D({\mathfrak g}))$$ be the set of all $$\lambda\in {\mathfrak g}^*$$ such that there exists a non-zero semi-invariant of $$U$$ (respectively $$D$$) relative to $$\lambda$$. $$\Lambda({\mathfrak g})$$ is a semigroup which need not be finitely generated and S. Montgomery [Proc. Am. Math. Soc. 83, 263–268 (1981; Zbl 0474.16003)] has shown that the group $$\Lambda_ D({\mathfrak g})$$ is isomorphic to Kharchenko’s group of $$X$$-inner automorphisms of $$U$$. It is shown that $$\Lambda_ D({\mathfrak g})$$ is the additive subgroup of $${\mathfrak g}^*$$ generated by $$\Lambda({\mathfrak g}).$$
Let $${\mathfrak g}_{\Lambda}$$ denote the intersection of $$\mathrm{ker}\,\lambda$$ for $$\lambda\in \Lambda ({\mathfrak g})$$. Then $${\mathfrak g}_{\Lambda}$$ is a characteristic ideal of $${\mathfrak g}$$ such that $$Z(U)\subset \mathrm{Sz}(U)\subset Z(U_{\Lambda})=\mathrm{Sz}(U_{\Lambda}),$$ where $$Z(R)$$ denotes the center of the ring $$R$$ and $$U_{\Lambda}$$ denotes the universal enveloping algebra of $${\mathfrak g}_{\Lambda}$$. Moreover $$\mathrm{Sz}(U)=Z(U_{\Lambda})$$ in case $$k$$ is algebraically closed and either $${\mathfrak g}$$ is almost algebraic or Frobenius.
A brief section deals with a construction of semi-invariants. Various characterizations for $$U$$ to be primitive are given in terms of semi-invariants, for example there exists a non-zero semi-invariant e which is contained in each non-zero prime ideal of $$U$$.

##### MSC:
 17B35 Universal enveloping (super)algebras 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras 16P40 Noetherian rings and modules (associative rings and algebras) 16Dxx Modules, bimodules and ideals in associative algebras 13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
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