On the cohomology of the Lie algebra \(L_2\)
type: Item
instance of: scholarly article
title: On the cohomology of the Lie algebra \(L_2\)
zbMATH Open document ID: 0857.17018
author: Alice Fialowski
published in: Pacific Journal of Mathematics
publication date: 1996-11-03T00:00:00Z
review text: Let \(W\) be the Lie algebra of polynomial vector fields in \(\mathbb C\). Then \(W\) has a basis \(e_k=x^{k+1}d/dx\), \(k\geq -1\) with multiplication \([e_i,e_j]= (j-i)e_{i+j}\). For \(k\geq 0\), let \(L_k\) denote the subalgebra of \(W\) generated by \(\{e_j\mid j\geq k\}\). The cohomology rings \(H^*(L_k)\), \(k\geq 0\), are known. For \(\lambda,\mu\in \mathbb C\), define the \(W\)-module \({\mathcal F}_{\lambda,\mu}\) to be the complex vector space with basis \(f_j\), \(j\in\mathbb Z\) and \(W\)-action defined by \(e_i\cdot f_j=(j+\mu-(i+1)\lambda)f_{i+j}\). The subspace \(F_{\lambda,\mu}\) with basis \(f_j\), \(j\geq 0\) and the same action is an \(L_0\)-module. The homology of \(L_1\) with coefficients in \({\mathcal F}_{\lambda,\mu}\) and \(F_{\lambda,\mu}\) has been previously computed by \textit{B. L. Feigin} and \textit{D. B. Fuchs} [Funct. Anal. Appl. 14, No. 3, 201--212 (1981); translation from Funkts. Anal. Prilozh. 14, No. 3, 45--60 (1980; Zbl 0482.57010)]. In this paper the author computes the homology in dimensions 0, 1, and 2 for the algebra \(L_2\) with coefficients in the modules \({\mathcal F}_{\lambda,\mu}\) and \(F_{\lambda,\mu}\) and offers conjectures, with supporting arguments, that the higher-dimensional homology is zero for all values of \(\lambda\) and \(\mu\).
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B66
zbMATH DE Number: 910265
zbMATH Keywords: Lie algebra \(L_2\)
zbMATH Keywords: Lie algebra of polynomial vector fields
zbMATH Keywords: cohomology
zbMATH Keywords: homology
reviewed by: Duncan J. Melville
MaRDI profile type: Publication
Wikidata QID: Q115230726
full work available at URL: pjm.1996.172.399
OpenAlex ID: W2592850913
DOI: 10.2140/PJM.1996.172.399
Recommended article: Stable cohomology of the algebra \(W_ n\) and relations of the algebra \(L_ 1\)
Recommended article: On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators
Recommended article: Homology of the Lie algebra of vector fields on the line with coefficients in symmetric powers of its adjoint representation
Recommended article: Homology groups of nilpotent subalgebras of the Lie superalgebra \(K(1,1)\). II
Recommended article: Cohomology theory of the Virasoro algebra
Recommended article: Q3812415
Recommended article: Stable cohomologies of Lie algebras of formal vector fields with tensor coefficients
Recommended article: A class of graded Lie algebras of vector fields and first order differential operators
Recommended article: Chevalley-Eilenberg algebraic cohomology space of polynomial vector fields Lie algebras on the line
Recommended article: Filtering bases and cohomology of nilpotent subalgebras of the Witt algebra and the algebra of loops in \(sl_2\)
description: scientific article; zbMATH DE number 910265