A new class of unitary representations for affine Lie algebras
type: Item
instance of: scholarly article
title: A new class of unitary representations for affine Lie algebras
zbMATH Open document ID: 0662.17009
DOI: 10.1016/0021-8693(89)90188-9
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: The author is concerned with constructing irreducible representations of affine Lie algebras which are unitarizable with respect to any form coming from a finite-dimensional Lie algebra. Let \(L(\mathfrak g) = \mathfrak g\otimes_{\mathbb C}\mathbb C[t,t^{-1}]\) be a loop algebra of a finite-dimensional simple Lie algebra \(\mathfrak g\) and \(\overline{L}(\mathfrak g)\) be a semi-direct product of \(L(\mathfrak g)\) and a derivation \(d\) defined by \([d, X\otimes t^ n]=n\,X\otimes t^ n\). Fix \(k\geq 1\) and let \(\mathfrak g_ k=\oplus_{k}\mathfrak g\) be a direct sum of \(k\)-copies of \(\mathfrak g\). Let \(\mathfrak a=(a_1,\ldots,a_k)\) be a sequence of distinct nonzero complex numbers. Define the Lie algebra homomorphism \(\Phi(\mathfrak a): L(\mathfrak g)\to\mathfrak g_ k\) by \(\Phi(\mathfrak a)(X\otimes t^n) = (a_1^nX,\ldots,a^n_kX)\), \(X\in\mathfrak g\), \(n\in\mathbb Z\). Let \((V,\tau)\) be a \(\mathfrak g_k\)-module and define an \(\overline{L}(\mathfrak g)\)-module \((L(V),\Pi(\mathfrak a))\) by \[ \Pi (\mathfrak a)(X\otimes t^n)(v\otimes t^m) = (\tau (\Phi (\mathfrak a)(X\otimes t^n))v)\otimes t^{n+m},\quad \Pi (\mathfrak a) dv\otimes t^m = mv\otimes t^ m \] for all \(X\in\mathfrak g\), \(v\in V\) and for all \(m,n\in\mathbb Z\). The author investigates the \(\overline{L}(\mathfrak g)\)-module \(L(V)\). Fix a Cartan subalgebra \(\mathfrak h\subset\mathfrak g\) and let \(\mathfrak h_k=\oplus_k \mathfrak h\). The author shows that if \((V,\tau)\) is an irreducible \(\mathfrak g_k\)-module which is also an \(\mathfrak h_k\)-weight module, then \((L(V),\Pi(\mathfrak a))\) is a completely reducible \(\overline{L}(\mathfrak g)\)-module, if further \(\mathfrak a=(a_1,\ldots,a_k)\) is chosen such that \((a_i/a_j)^{k!}\neq 1\) for \(i\neq j\), then \((L(V),\Pi(\mathfrak a))\) is an irreducible \(\overline{L}(\mathfrak g)\)-module and he gives a counter-example to show that it is not always irreducible. Further, he shows that \(L(V)\) is unitarizable with respe…
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 4081783
zbMATH Keywords: unitarizable representations
zbMATH Keywords: root
zbMATH Keywords: highest weight
zbMATH Keywords: affine Lie algebras
Wikidata QID: Q115366164
author: S. Eswara Rao
reviewed by: Eiichi Abe
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90188-9
OpenAlex ID: W2115859244
cites work: New unitary representations of loop groups
cites work: Q3693664
description: scientific article; zbMATH DE number 4081783