Central extensions of Lie superalgebras
type: Item
instance of: scholarly article
title: Central extensions of Lie superalgebras
zbMATH Open document ID: 1036.17004
author: Kenji Iohara
author: Yoshiyuki Koga
published in: Commentarii Mathematici Helvetici
publication date: 2001-01-01T00:00:00Z
reviewed by: Vyacheslav A. Artamonov
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 17B55
zbMATH DE Number: 1610944
zbMATH Keywords: Lie superalgebras
MaRDI profile type: Publication
full work available at URL: s000140050152
OpenAlex ID: W2031411187
DOI: 10.1007/S000140050152
review text: A finite dimensional Lie superalgebra \(\mathfrak g=\mathfrak g_0\oplus \mathfrak g_1\) is basic classical ifNEWLINENEWLINE(1) \(\mathfrak g\) is a simple Lie superalgebra;NEWLINENEWLINE(2) \(\mathfrak g_0\) is a reductive Lie algebra;NEWLINENEWLINE (3) there exists a non-degenerate bilinear form \(b(x,y)\) on \(\mathfrak g\) such thatNEWLINENEWLINEa) \(\mathfrak g_0,\,\mathfrak g_1\) are orthogonal with respect to \(b(x,y)\);NEWLINENEWLINEb) \(b(x,y)=(-1)^{| x| \,| y| }b(y,x)\) for all homogeneous elements \(x,y\in \mathfrak g\);NEWLINENEWLINEc) \(b([a,b],c)=b(a,[b,c])\) for all \(a,b,c\in \mathfrak g\).NEWLINENEWLINE Suppose that \(\mathfrak g\) is a complex basic classical Lie superalgebra such that if \(\mathfrak g\) is of a type \(D(2,1;a)\) then \(a\in \mathbb Z\setminus \{0,\,-1\}\). It is shown that there exists a \(\mathbb Z\)-form \(\mathfrak g_{\mathbb Z}\) of \(\mathfrak g\).NEWLINENEWLINE For any commutative ring \(k\) put \(g_k=k\otimes_{\mathbb Z}\mathfrak g_{\mathbb Z}\). Suppose that \(A\) is a commutative \(k\)-algebra which is a free \(k\)-module andNEWLINENEWLINE(i) 2 is invertible in \(k\) if \(\mathfrak g\) is of one of types \(A(m,n),\, C(n),\, D(m,n)\);NEWLINENEWLINE(ii) 6 is invertible in \(k\) if \(\mathfrak g\) is of types \(B(m,n),\, F(4),\, G(3)\);NEWLINENEWLINE(iii) \(2a(a+1)\) is invertible in \(k\) if \(\mathfrak g\) is of type \(D(2,1;a)\).NEWLINENEWLINEThen the universal central extension \(\mathfrak g_k(A)\) of \(\mathfrak g_k\otimes A\) has the form NEWLINE\[NEWLINE \mathfrak g_k(A)= \begin{cases} \mathfrak g_k \otimes A\oplus \Omega^1_{A/k}/dA , \quad & \mathfrak g \text{ is not of type } A(n,n), \\ \mathfrak d_k \otimes A\oplus \Omega^1_{A/k}/dA , \quad & \mathfrak g \text{ is of type } A(1,1),\\ \mathfrak(n+1,n+1)_k\oplus \Omega^1_{A/ k}/dA , \quad & \mathfrak g \text{ is of type } A(n,n),\, n>1.\end{cases} NEWLINE\]NEWLINE Here \(\Omega^1_{A/k}/dA\) is the \(A\)-module of Kähler differentials and \(\mathfrak d\) the Lie supera…
description: scientific article; zbMATH DE number 1610944