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On the global dimension of certain primitive factors of the enveloping algebra of a semi-simple Lie algebra. (English) Zbl 0588.17009
Let $${\mathfrak g}={\mathfrak n}^++{\mathfrak h}+{\mathfrak n}^-$$ be a triangular decomposition of the semisimple Lie algebra $${\mathfrak g}$$ over an algebraically closed field of characteristic zero. The authors are interested in obtaining information concerning the global dimension of the primitive factor rings $$D_{\lambda}=U({\mathfrak g})/ann M(\lambda)$$, for Verma modules M($$\lambda)$$. In the case that $$\lambda$$ is regular they obtain an upper bound in terms of the dimension of $${\mathfrak n}^+$$. Once it is know that gl. dim(D$${}_{\lambda})$$ is finite earlier work of Levasseur can be used to give an exact formula for the global dimension.
In preparation for these results the authors obtain a torsion theoretic localisation result for the global dimension of a Noetherian prime ring that is of independent interest. Let R be a Noetherian prime ring with quotient ring Q and let $$B_ 1,...,B_ n$$ be rings lying between R and Q and such that (i) $$B_ 1\oplus... \oplus B_ n$$ is faithfully flat over R and (ii) $$B_ i\otimes_ R B_ j\cong B_ j\otimes_ R B_ i$$ as R-R bimodules. Then gl. dim(R)$$\leq \max \{gl. \dim (B_ i)+flat \dim (B_ i)\}$$. The interesting point here is that in contrast to earlier results of this type there is no initial assumption that gl. dim(R) is finite. In order to use this result to obtain the result mentioned earlier the authors employ the Conze embedding of $$D_{\lambda}$$ into the Weyl algebra $$A_ n$$, where $$n=\dim ({\mathfrak n}^+)$$.
Reviewer: T.H.Lenagan

##### MSC:
 17B35 Universal enveloping (super)algebras 16E10 Homological dimension in associative algebras 16S90 Torsion theories; radicals on module categories (associative algebraic aspects) 17B20 Simple, semisimple, reductive (super)algebras 16P40 Noetherian rings and modules (associative rings and algebras)
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