Quantised Function Algebras at Roots of Unity and Path Algebras

Let G be a semisimple algebraic group and ε a primitive lth root of unity. In this paper we realise certain restricted quantised enveloping algebras of Borel subalgebras of semisimple Lie algebras and the analogue of these restricted enveloping algebras for Oε[G] as path algebras of certain quivers. This extends several results in Cibils (Internat. Math. Res. Notices12 (1997), 541–553). As an application of this description we describe the extensions of simple modules of the quotient of Oε[G].