Complexity of Representations of Quantised Function Algebras and Representation Type

Let G be a simply connected, connected, semisimple algebraic group over , B± be opposite Borel subgroups of G, and = Lie(B+ ). Let Oε[G] be the quantised function algebra at a root of unity ε and let U≥ 0ε be the quantised enveloping algebra of at a root of unity. We study the finite dimensional factor algebras Oε[G](g) and U≥ 0ε(b) for g G and b B− , which were introduced by De Concini, Kac, and Procesi (1992, in “Geometry and Analysis,” pp. 41–65, Tata Inst. Fund. Res., Bombay) and De Concini and Lyubashenko (1994, Adv. Math.108, 205–262). In particular we describe the complexity of these factor algebras in terms of the Weyl group of G and deduce results on their representation type. Finally we completely describe those factor algebras of finite representation type.