The Ramifications of the Centres: Quantised Function Algebras at Roots of Unity

This paper continues the study of quantised function algebras Oǫ[G] of a semisimple group G at an lth root of unity ǫ. These algebras were introduced by De Concini and Lyubashenko in 1994, and studied further by De Concini and Procesi and by Gordon, amongst others. Our main purpose here is to increase understanding of the finite dimensional factor algebras Oǫ[G](g), for g ∈ G. We determine the representation type and block structure of these factors, and (for many g) describe them up to isomorphism. A series of parallel results is obtained for the quantised Borel algebras U ǫ and U ≤0 ǫ .