Cohomology of nilpotent subalgebras of the algebra \(\widehat{sl_2}\)
type: Item
instance of: scholarly article
title: Cohomology of nilpotent subalgebras of the algebra \(\widehat{sl_2}\)
zbMATH Open document ID: 0901.17014
DOI: 10.1007/BF02509626
publication date: 1997-11-05T00:00:00Z
review text: The algebra of polynomial \(sl_2\)-currents on the line is considered, and the nilpotent subalgebras of the type \[ N(P_+, P_0, P_-):= P_+(t)\mathbb{C} [t]\otimes e+P_0(t) \mathbb{C}[t]\otimes h+P_-(f)\cdot \mathbb{C}[t]\otimes f \] are the main object of study, where \(P_0, P_+, P_-\) are polynomials such that \(P_0\) divides \(P_+\cdot P_-\) and \(e,h,f\) are the standard generators of \(sl_2\). The general result, Theorem 1, is if \(\text{deg }P_++ \text{deg }P_->0\) then the dimension of the homology \(H_i(N(P_+, P_0, P_-))\) depends only on the degrees of \(P_0, P_+, P_-\). Actually the dimensions are computed. The next two theorems deal with specific polynomials. In Theorem 3 the bigraded space structure in \(H_i(N (t^{h_+}, t^{h_0}, t^{h_-}))\) is computed, and in Theorem 2 the case when \(P_0, P_+, P_-\) has no multiple roots is worked out in more detail. The authors point out the difficulty they have met when trying to replace \(sl_2\) by another semisimple Lie algebra.
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B65
zbMATH DE Number: 1053628
zbMATH Keywords: current Lie algebras
zbMATH Keywords: cohomologies
zbMATH Keywords: polynomial \(sl_2\)-currents
zbMATH Keywords: nilpotent subalgebras
author: B. Feigin
author: Sergey A. Loktev
reviewed by: Alexei Rudakov
MaRDI profile type: Publication
cites work: Cohomology of Lie algebras of formal vector fields on the line
cites work: Q3819212
full work available at URL: https://doi.org/10.1007/bf02509626
OpenAlex ID: W2072384560
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 1053628