On the K -theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case

Patrick Polo

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 3, page 707-720
  • ISSN: 0373-0956

Abstract

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Let G be a semisimple complex algebraic group and X its flag variety. Let 𝔤 = Lie ( G ) and let U be its enveloping algebra. Let 𝔥 be a Cartan subalgebra of 𝔤 . For μ ∈ 𝔥 * , let J μ be the corresponding minimal primitive ideal, let U μ = U / J μ , and let 𝒯 U μ : K 0 ( U m u ) → ℂ 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 ℂ -algebras U μ . When μ is regular, Hodges has shown that K 0 ( U μ ) ≅ K 0 ( X ) . In this case K 0 ( U μ ) is generated by the classes corresponding to G -linearized line bundles on X , and the value of 𝒯 U μ on these generators was computed by Hodges and Holland, in a special case, and then by Perets and the author, in general. This result is extended here to the singular case.

How to cite

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Polo, Patrick. "On the $K$-theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case." Annales de l'institut Fourier 45.3 (1995): 707-720. <http://eudml.org/doc/75135>.

@article{Polo1995,
abstract = {Let $G$ be a semisimple complex algebraic group and $X$ its flag variety. Let $\{\frak g\}=\{\rm Lie\}(G)$ and let $U$ be its enveloping algebra. Let $\{\frak h\}$ be a Cartan subalgebra of $\{\frak g\}$. For $\mu \in \{\frak h\}^*$, let $J_\mu $ be the corresponding minimal primitive ideal, let $U_\mu =U/J_\mu $, and let $\{\cal T\}_\{U_\mu \}:K_0(U_mu)\rightarrow \{\Bbb C\}$ 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 $\{\Bbb C\}$-algebras $U_\mu $. When $\mu $ is regular, Hodges has shown that $K_0(U_\mu )\cong K_0(X)$. In this case $K_0(U_\mu )$ is generated by the classes corresponding to $G$-linearized line bundles on $X$, and the value of $\{\cal T\}_\{U_\mu \}$ on these generators was computed by Hodges and Holland, in a special case, and then by Perets and the author, in general. This result is extended here to the singular case.},
author = {Polo, Patrick},
journal = {Annales de l'institut Fourier},
keywords = {Hattori-Stallings trace; enveloping algebras; semisimple Lie algebras; minimal primitive ideal},
language = {eng},
number = {3},
pages = {707-720},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the $K$-theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case},
url = {http://eudml.org/doc/75135},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Polo, Patrick
TI - On the $K$-theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 3
SP - 707
EP - 720
AB - Let $G$ be a semisimple complex algebraic group and $X$ its flag variety. Let ${\frak g}={\rm Lie}(G)$ and let $U$ be its enveloping algebra. Let ${\frak h}$ be a Cartan subalgebra of ${\frak g}$. For $\mu \in {\frak h}^*$, let $J_\mu $ be the corresponding minimal primitive ideal, let $U_\mu =U/J_\mu $, and let ${\cal T}_{U_\mu }:K_0(U_mu)\rightarrow {\Bbb C}$ 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 ${\Bbb C}$-algebras $U_\mu $. When $\mu $ is regular, Hodges has shown that $K_0(U_\mu )\cong K_0(X)$. In this case $K_0(U_\mu )$ is generated by the classes corresponding to $G$-linearized line bundles on $X$, and the value of ${\cal T}_{U_\mu }$ on these generators was computed by Hodges and Holland, in a special case, and then by Perets and the author, in general. This result is extended here to the singular case.
LA - eng
KW - Hattori-Stallings trace; enveloping algebras; semisimple Lie algebras; minimal primitive ideal
UR - http://eudml.org/doc/75135
ER -

References

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