Representations of quantum groups at even roots of unity
Let \(U_ q({\mathfrak g})\) be the quantum enveloping algebra of the finite dimensional complex simple Lie algebra \({\mathfrak g}\) where \(q\) is a primitive \(\ell\)-th root of unity. When \(\ell\) is odd, the representation theory of \(\text{Spec } U_ q({\mathfrak g})\), the finite-dimensional irreducible representations was determined by \textit{C. De Concini} and \textit{V. Kac} [Colloq. Dixmier, Prog. Math. 92, 471-506 (1990; Zbl 0738.17008)], who partitioned \(\text{Spec } U_ q({\mathfrak g})\) by conjugacy classes of the corresponding algebraic group \(G\). The author of the paper under review handles the case when \(\ell\) is even, especially when \(\ell\) is divisible by 4. Here the partition goes by conjugacy classes in the group \(G^ \vee\), the Langlands dual of \(G\), and is more complicated than in the odd case. There is a discussion of a quantum coadjoint action and triangulability, and the degree of \(U_ q({\mathfrak g})\) is calculated by a deformation argument.
scientific article; zbMATH DE number 614197