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Zbl 0818.17006
Polo, Patrick
On the $K$-theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras: the singular case. (English)
[J] Ann. Inst. Fourier 45, No. 3, 707-720 (1995). ISSN 0373-0956; ISSN 1777-5310/e

Summary: Let $G$ be a semisimple complex algebraic group and $X$ its flag variety. Let ${\germ g}=\text{Lie}(G)$ and let $U$ be its enveloping algebra. Let ${\germ h}$ be a Cartan subalgebra of ${\germ g}$. For $\mu\in{\germ h}\sp*$, let $J\sb \mu$ be the corresponding minimal primitive ideal, let $U\sb \mu=U\slash J\sb \mu$, and let ${\cal T}\sb{U\sb \mu}:K\sb 0(U\sb \mu)\to{\bbfC}$ be the Hattori-Stallings trace. Results of Hodges suggest to study this map as a step towards a classification, up to isomorphism or Morita equivalence, of the ${\bbfC}$-algebras $U\sb \mu$. When $\mu$ is regular, Hodges has shown that $K\sb 0(U\sb \mu)\cong K\sb 0(X)$. In this case $K\sb 0(U\sb \mu)$ is generated by the classes corresponding to $G$-linearized line bundles on $X$, and the value of ${\cal T}\sb{U\sb \mu}$ on these generators was computed by Hodges and Holland, in a special case, and then by G. S. Perets and the author, in general. This result is extended here to the singular case.

MSC 2000:: *17B35 Universal enveloping algebras (Lie algebras)
16E20 K-theory of noncommutative rings
16S30 Universal enveloping algebras of Lie algebras (associative)
14M15 Grassmannians, Schubert varieties

Keywords: Hattori-Stallings trace; enveloping algebras; semisimple Lie algebras; minimal primitive ideal

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