Ringel-hall algebras and two-parameter quantized enveloping algebras

Document Type

Article

Publication Date

11-1-2010

Abstract

Let g be a finite-dimensional complex simple Lie algebra and 3 be the finite dimensional hereditary algebra associated to g. Let U+r,s (g) (respectively U? 0r,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? 0r,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 Hr,s (?), which generalizes a result of Reineke. Based on detailed computations in Hr,s (?), we show that Hr,s (?) can be presented as an iterated skew polynomial ring. As an result, we obtain a PBW-basis for Hr,s (?), 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 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. In particular, we define a Hopf algebra structure on; and we prove that U? 0r,s (g) is isomorphic as a Hopf algebra to the two-parameter extended Ringel-Hall algebra. © 2010 by Pacific Journal of Mathematics.

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