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 massive strengthen 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 performed, this paintings exhibits that many partial effects can be found.

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**Example text**

Fix a Borel subalgebra b of g and a Cartan subalgebra f) contained in b. Choose a coset A of the weight lattice P of 0 in rj* and let A + + denote its intersection with the dominant (open) Weyl chamber (with respect to the integral root system of any representative of A, not the full root system of Q). Denote the Weyl group of Q by W and write W\ = W\ to denote the subgroup of W generated by the integral reflections on A, for any A in A. X = w(X + p) — p, where p as usual denotes the half sum of the positive roots relative to b (cf.

Now Lusztig gives a recipe for reading off the order of A{U) from the symbol of q in [36, chapter 4]. One easily checks that this recipe is equivalent to the one given in the statement of the theorem. D. Now we want to explore the relationship between a g-unipotent Dixmier algebra and its associated variety more deeply. To do this it is useful to observe that we can apply all the results of Chapter 3 to g-unipotent Dixmier algebras. 5. Let M be an irreducible Harish-Chandra module for U(#). We say that M is parabolically weakly unipotent if whenever g is isomorphic to a derived Levi factor [[, I] of a larger semisimple algebra gf and E is a finitedimensional i-module with highest weight in the root lattice ofg', then M~~E has no irreducible subquotients of infinitesimal character strictly shorter than that of M. ~~

6, relative to A^, A^, respectively. X'i) are all induced as Dixmier algebras. 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/!