Degree one cohomology for the Lie algebras of derivations
type: Item
instance of: scholarly article
title: Degree one cohomology for the Lie algebras of derivations
zbMATH Open document ID: 1044.17012
author: Serge Skryabin
published in: Lobachevskii Journal of Mathematics
publication date: 2004-02-10T00:00:00Z
full work available at URL: https://eudml.org/doc/233525
full work available at URL: skr.html
review text: Cohomology of Lie algebras of vector fields is also known as Gelfand-Fuks cohomology owing to \textit{I. M. Gelfand} and \textit{D. B. Fuks} [Funkts. Anal. Prilozh. 3, No. 2, 87 (1969; Zbl 0194.24601), Funct. Anal. Appl. 3, 194--210 (1970); translation from Funkts. Anal. Prilozh. 3, No. 3, 32--52 (1969; Zbl 0216.20301)] who introduced cochain complexes of continuous cochains in order to remove infinite dimensionality of (some) cohomology spaces. The article under review proposes a new setup: the object of study are Lie subalgebras \(W\) of the Lie algebra of derivations of a commutative associative unital ring \(R\), forming a projective module of constant finite rank \(n\geq 1\) under the natural \(R\)-module structure. While the traditional class of examples are Lie algebras of smooth vector fields \(\text{Vect}(X)\) on \({\mathcal C}^{\infty}\) manifolds \(X\), holomorphic fields on Stein manifolds or algebraic fields on smooth algebraic varieties and their respective subalgebras, the generality of the present approach allows to include for example Cartan type Lie algebras in positive characteristic (standing assumption is however \(2^{-1}\in R\)). Amazingly, the author proves results on derivations and central extensions for \(W\), i.e. for degree \(1\) cohomology with values in some category of representations of \(W\) (a subcategory of \({\mathcal C}_1\) to be defined below) and degree \(2\) cohomology with trivial coefficients, using only algebraic (i.e. possibly non-continuous) cochains (in fact, the author works over \({\mathbb Z}\) instead of a field). This allows to take on cohomology of vector fields a nice algebraic point of view, though it seems to be not suited for higher degree cohomology. The theorem on central extensions reads as follows: [Theorem 7.1] Assume that \(3R=R\). If \(\text{rk}_RW>1\), then every \({\mathbb Z}\)-split central extension of \(W\) splits. If \(\text{rk}_RW=1\), then the universal central extension of \(W\) has kernel \(H^1(…
reviewed by: Friedrich Wagemann
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B66
Mathematics Subject Classification ID: 13C10
zbMATH DE Number: 2039394
zbMATH Keywords: degree one cohomology
zbMATH Keywords: Lie algebras of derivations
zbMATH Keywords: (universal) central extensions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2039394