On tensor categories of Lie type \(E_{N}, N\neq 9\)
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.
scientific article; zbMATH DE number 1957033