Ringel-Hall algebras and two-parameter quantized enveloping algebras
Let \(\Lambda\) be the finite-dimensional hereditary algebra associated with a finite-dimensional complex simple Lie algebra \(\mathfrak g\). In the paper the two-parameter quantized enveloping algebras \(U_{r,s}^+(\mathfrak g)\) and \(U_{r,s}^{\geq 0}(\mathfrak g)\) are studied, where \(U_{r,s}^+(\mathfrak g)\) (resp. \(U_{r,s}^{\geq 0}(\mathfrak g)\)) is the two-parameter quantized enveloping algebra of the positive maximal nilpotent (resp. Borel) Lie subalgebra of \(\mathfrak g\). These algebras are investigated using the approach of Ringel-Hall algebras. It is proved that \(U_{r,s}^+(\mathfrak g)\) is isomorphic to a certain two-parameter twisted Ringel-Hall algebra \(H_{r,s}(\Lambda)\). This is a generalisation of a result of \textit{M. Reineke} [Represent. Theory 5, 147--163 (2001; Zbl 1050.17015)], that proved these isomorphisms for complex simple Lie algebras of types \(\mathbb{A,D,E}\). Applying computations in \(H_{r,s}(\Lambda)\)), the author showed that \(H_{r,s}(\Lambda)\) can be presented as an iterated skew polynomial ring. As a consequence of this presentation, a PBW-basis for \(H_{r,s}(\Lambda)\) is obtained. It is also proved that all prime ideals of \(U_{r,s}^+(\mathfrak g)\) are completely prime under some mild conditions on the parameters \(r,s\). Moreover, the author studies the two-parameter extended Ringel-Hall algebra \(\overline{H_{r,s}(\Lambda)}\) (by adding the torus part to it). In particular, a Hopf algebra structure on \(\overline{H_{r,s}(\Lambda)}\) is defined; and it is proved that \(U_{r,s}^{\geq 0}(\mathfrak g)\) is isomorphic as a Hopf algebra to the two-parameter extended Ringel-Hall algebra \(\overline{H_{r,s}(\Lambda)}\).
two-parameter quantized enveloping algebras
scientific article; zbMATH DE number 5809036