# Completely Prime Maximal Ideals and Quantization by William M. McGovern

By William M. McGovern

This monograph will attract graduate scholars and researchers drawn to Lie algebras. McGovern classifies the thoroughly leading maximal spectrum of the enveloping algebra of any classical semisimple Lie algebra. He additionally stories finite algebra extensions of thoroughly leading primitive quotients of such enveloping algebras and computes their lengths as bimodules, attribute cycles, and Goldie ranks in lots of situations. This paintings marks a tremendous boost within the quantization software, which seeks to increase the equipment of (commutative) algebraic geometry to the surroundings of enveloping algebras. whereas such an extension can't be thoroughly conducted, this paintings exhibits that many partial effects can be found.

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**Example text**

At this point we assume that g is of type B. We will give a similar argument in type D in a moment and then a very different one for type C. Consider now infinitesimal characters A obtained by starting with a basic character and then adding a 1 if it has only half-integer coordinates, or | if it has only integer coordinates. Such characters may be paired off as follows. If the "basic part" A^ of one of them has integer coordinates, it corresponds to a certain partition of an odd integer 2ra + 1 having only odd terms.

Ll)) and let A be an algebra in U\. The length of A as a bimodule is given as follows. Let p be the partition of the sl(n) orbit corresponding to A. (a) in type A, this length is 1. (b) in type D, break up q := p0dd into chunks in the following way. First write q as ( g 1 1 , . . ,

The first two are induced from algebras with basic infinitesimal character, so that one can compute their Goldie ranks as in the first paragraph of the proof. 19. 15(b) extended to A^ and A^. Xh) for any y in the integral Weyl group. If Q is of type £>, let A^ be a typical infinitesimal character obtained from 42 WILLIAM M. MCGOVERN a basic one with integer coordinates by adding ( | , | ) . 5, and then adding (1,0). Once more we have #(C/! fl C^1) = 1. Xi) and A(wh-Xh). Replace A^, Xh by their basic parts and then tensor by (§, | , | , 5), ( 1 , - 1 , 3 , 1 ) , respectively.