Certain topics on the representation theory of the Lie algebra \(gl(\lambda)\)
type: Item
instance of: scholarly article
title: Certain topics on the representation theory of the Lie algebra \(gl(\lambda)\)
zbMATH Open document ID: 0916.17019
DOI: 10.1007/BF02434006
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-03-25T00:00:00Z
review text: The paper surveys recent results in the representation theory of the Lie algebra \(g=gl(\lambda)\) and some applications to combinatorial identities. The Lie algebra \(g\), where \(\lambda\) is a complex parameter, is a continuous version of the Lie algebra \(gl_{\infty}\) and arises from the homology ring of the Lie algebra of differential operators on the line. It is \({\mathbb Z}\)-graded and possesses an analogue of the Cartan decomposition. It is known that the problem of irreducibility of \(g\)-modules can be reduced to the corresponding problem for \(\widehat{gl}_{\infty}\)-modules. In Chapter 1 the author describes the structure of \(\widehat{gl}_{\infty}\)-modules induced from the largest parabolic subalgebra. In particular, he describes the Jantzen filtration and computes the determinant of the Shapovalov form. As a corollary, one obtains a character formula for the first consecutive module of the Jantzen filtration, which can be applied to get the classical and ``higher'' Euler identities. The subject of Chapter 2 is the representation theory of \(g\). The author describes embeddings into \(\widehat{gl}_{\infty}\) and deduces a criterion for reducibility of those representations of \(g\) that are induced from the special parabolic subalgebra. This is applied to obtain certain identities called ``local''. Finally, in Chapter 3 the author considers the Lie algebra of functions on a hyperboloid, which can be flat deformed in \(g\). Then the author applies all previous results to obtain a new combinatorial identity, which is called ``global''.
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 17B66
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 17-02
zbMATH DE Number: 1269458
zbMATH Keywords: combinatorial identities
zbMATH Keywords: infinite-dimensional Lie algebra
zbMATH Keywords: induced representation
zbMATH Keywords: Shapovalov form
zbMATH Keywords: irreducible representation
zbMATH Keywords: Jantzen filtration
author: Boris Shoikhet
MaRDI profile type: Publication
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Wikidata QID: Q115392190
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description: scientific article; zbMATH DE number 1269458