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The linkage principle for restricted critical level representations of affine Kac–Moody algebras

Part of: Lie algebras and Lie superalgebras Groups and algebras in quantum theory

Published online by Cambridge University Press:  11 October 2012

Tomoyuki Arakawa  and
Peter Fiebig
Tomoyuki Arakawa
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan (email: arakawa@kurims.kyoto-u.ac.jp)
Peter Fiebig
Affiliation:
Department Mathematik, Emmy-Noether-Zentrum, FAU Erlangen-Nürnberg, Cauerstr. 11, D-91058, Germany (email: fiebig@mi.uni-erlangen.de)
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Abstract

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We study the restricted category 𝒪 for an affine Kac–Moody algebra at the critical level. In particular, we prove the first part of the Feigin–Frenkel conjecture: the linkage principle for restricted Verma modules. Moreover, we prove a version of the Bernstein–Gelfand–Gelfand-reciprocity principle and we determine the block decomposition of the restricted category 𝒪. For the proofs, we need a deformed version of the classical structures, so we mostly work in a relative setting.

Keywords

linkage principle

MSC classification

Secondary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 81R10: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current algebras and their representations
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1787 - 1810
Copyright
Copyright © Foundation Compositio Mathematica 2012

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