Durch graduierte Lie-Algebren definierte Paar-Algebren (Pair algebras defined by graded Lie algebras)
type: Item
instance of: scholarly article
title: Durch graduierte Lie-Algebren definierte Paar-Algebren (Pair algebras defined by graded Lie algebras)
zbMATH Open document ID: 0569.17006
DOI: 10.1007/BF01171488
author: Wolfgang Hein
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155080
review text: A pair \({\mathcal A}=(A^-,A^+)\) of vector spaces over a field K with trilinear mappings (x,u,y)\(\to \{xuy\}_{\epsilon}\) from \(A^{\epsilon}\times A^{-\epsilon}\times A^{\epsilon}\) into \(A^{\epsilon}\) is called a pair algebra if \(\{xu\{yvz\}_{\epsilon}\}_{\epsilon}- \{yv\{xuz\}_{\epsilon}\}_{\epsilon}\}=\{\{xuy\}_{\epsilon}vz\}_{\epsilon}- \{y\{uxv\}_{-\epsilon}z\}_{\epsilon}\) for all \(x,y,z\in A^{\epsilon}\), \(u,v\in A^{-\epsilon},\epsilon =\pm\). With a pair algebra \({\mathcal A}\) there is associated a certain graded Lie algebra \({\mathcal S}({\mathcal A})=\oplus S({\mathcal A})_ r\) (r\(\in {\mathbb{Z}})\), known as the standard algebra of A, in which S(\({\mathcal A})_{-1}=A^-\) and S(\({\mathcal A})_ 1=A^+\). The author's investigation includes a characterization of those pair algebras A for which S(\({\mathcal A})\) is a Kac-Moody algebra as being pair algebras having a nondegenerate (left- and right-) invariant bilinear form and for which the inner derivation algebra is completely reducible. More detailed results are obtained when S(\({\mathcal A})\) is an affine Lie algebra.
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 3908642
zbMATH Keywords: graded Lie algebra
zbMATH Keywords: Kac-Moody algebra
zbMATH Keywords: pair algebras
zbMATH Keywords: invariant bilinear form
zbMATH Keywords: affine Lie algebra
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3908642