Complexity of representations of quantised function algebras and representation type
Let \(G\) be a complex simply connected, connected, semisimple algebraic group, \(B^+\) and \(B^-\) opposite Borel subgroups of \(G\), \({\mathfrak b}\) the Lie algebra of \(B^+\), \(O_\varepsilon [G]\) the quantized function algebra of \(G\) at a root \(\varepsilon\) of unity and \(U_\varepsilon\) the quantized enveloping algebra of \({\mathfrak b}\). The author studies some finite-dimensional algebras related to the latter two quantized algebras. These finite-dimensional algebras were also involved in the work of \textit{C. De Concini} and \textit{V. Lyubashenko} [Adv. Math. 108, 205-262 (1994; Zbl 0846.17008)] and of \textit{C. de Concini, V. G. Kac} and \textit{C. Procesi} [Geometry and Analysis, Tata Inst. Fundam. Res. 13, 41-65 (1995; Zbl 0878.17014)]. The author describes the complexity of these finite-dimensional algebras in terms of the Weyl group of \(G\). (The complexity is a measure of the rate of growth of minimal projective resolutions of finite-dimensional modules over the algebra.) He uses these complexity results to deduce results on the representation types of the algebras. He gives a complete description of those algebras which are of finite representation type.
scientific article; zbMATH DE number 1556965
scientific article