Ringel–Hall algebras and two-parameter quantized enveloping algebras

Abstract

Let g be a finite-dimensional complex simple Lie algebra and A be the finite-dimensional hereditary algebra associated to g. Let U + r,s (g) (respectively U ≥0 r,s (g)) denote the two-parameter quantized enveloping algebra of the positive maximal nilpotent (respectively Borel) Lie subalgebra of g. We study the two-parameter quantized enveloping algebras U + r,s (g) and U ≥0 r,s (g) using the approach of Ringel-Hall algebras. First of all, we show that U + r,s (g) is isomorphic to a certain two-parameter twisted Ringel-Hall algebra H r,s (A), which generalizes a result of Reineke. Based on detailed computations in H r,s (Λ), we show that H r,s (Λ) can be presented as an iterated skew polynomial ring. As an result, we obtain a PBW-basis for H r,s (A), which can be further used to construct a PBW-basis for the two-parameter quantized enveloping algebra U r,s (g). We also show that all prime ideals of U + r,s (g) are completely prime under some mild conditions on the parameters r, s. Second, we study the two-parameter extended Ringel-Hall algebra H r,s (Λ). In particular, we define a Hopf algebra structure on H r,s (Λ); and we prove that U ≥0 r,s (g) is isomorphic as a Hopf algebra to the two-parameter extended Ringel-Hall algebra H r,s (Λ).