Hopf structures on ambiskew polynomial rings

Abstract

We derive necessary and sufficient conditions for an ambiskew polynomial ring to have a Hopf algebra structure of a certain type. This construction generalizes many known Hopf algebras, for example U (sl 2 ), U q (sl 2 ) and the enveloping algebra of the three-dimensional Heisenberg Lie algebra. In a torsion-free case we describe the finite-dimensional simple modules, in particular their dimensions, and prove a Clebsch-Gordan decomposition theorem for the tensor product of two simple modules. We construct a Casimir type operator and prove that any finite-dimensional weight module is semisimple.

Ambiskew Hopf algebras

“…Special case: R commutative. When Theorem (2.4) is specialised to the case where R is commutative we obtain the main result of [Har08], albeit stated somewhat differently from there:…”

“…This is Theorem 2.4, with the rest of §2 containing the proof. The case where R is affine commutative over the algebraically closed field k was obtained previously as the main result of [Har08]. Even in the commutative case, however, we believe that the formulation given here makes it easier to determine the possible ambiskew extensions of a given Hopf algebra R.…”

“…Many interesting examples of ambiskew Hopf algebras with R commutative are given in [9,Section 4]. See also 4.1 and 4.2 below.…”

ON REPRESENTATIONS OF QUANTUM GROUPS Uq(fm(K,H))

“…2 REMARK 4.7. After the first draft of this paper was written, we have been kindly informed by J. Hartwig that the complete reducibility of finite dimensional weight representations is also proved in his preprint [6] in a more general setting of ambiskew polynomial rings via a different approach.…”

Construct irreducible representations of quantum groups U q (ƒ m (K))

“…✷ Remark 3.1. After the first draft of this paper was written, we have been kindly informed by J. Hartwig that the complete reducibility of finite dimensional weight representations is also proved in his preprint [6] in a more general setting of Ambiskew polynomial rings via a different approach. Remark 3.2.…”