On tensor categories of Lie type \(E_{N}, N\neq 9\)
type: Item
instance of: scholarly article
title: On tensor categories of Lie type \(E_{N}, N\neq 9\)
zbMATH Open document ID: 1038.17009
DOI: 10.1016/S0001-8708(02)00048-8
published in: Advances in Mathematics
publication date: 2003-07-30T00:00:00Z
review text: Let \({\mathfrak g}=\text{sl}_{N}(\mathbb{C})\) denote the Lie algebra of type \(A_{N-1}\). Let \(V\) denote the natural \(L\)-module of dimension \(N\). Then \({\mathfrak g}\) acts naturally on \(V^{\otimes n}\) for any \(n\) and the centraliser algebra is known as the Schur algebra corresponding to \(N\) and \(n\). In this paper, the authors study a similar situation in type \(E\). The Dynkin diagrams \(E_6\), \(E_7\) and \(E_8\) form part of an infinite series \(E_N\), where the longest arm of the graph is extended. Suppose now that \({\mathfrak g}={\mathfrak g}(E_N)\) is the corresponding Lie algebra (in general, a Kac-Moody Lie algebra). Let \(V\) be the fundamental representation of \({\mathfrak g}\) corresponding to the node in the diagram furthest from the branch point. If \(N=6\) or \(7\) then \(V\) is miniscule. The authors show that the centraliser algebra of the quantum group action (on the quantized version of \(V^{\otimes n}\)) is generated by the \(R\)-matrices together with one additional element. They show that an analogous statement holds in general for type \(E_N\), \(N>5\), provided \(E\not=8\), if \(V^{\otimes n}\) is replaced by an appropriate direct summand. In this way they obtain a \(2\)-parameter family of braid representations generalizing a \(q\)-verson of the Brauer centraliser algebras.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 1957033
zbMATH Keywords: semisimple Lie algebra
zbMATH Keywords: Kac-Moody Lie algebra
zbMATH Keywords: type \(E_N\)
zbMATH Keywords: quantum group
zbMATH Keywords: quantized enveloping algebra
zbMATH Keywords: tensor category
zbMATH Keywords: Littelmann path
zbMATH Keywords: braid representation
zbMATH Keywords: Brauer centraliser algebra
zbMATH Keywords: \(R\)-matrix.
author: Hans Wenzl
reviewed by: Bethany R. Marsh
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1957033