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