Finite-Dimensional Representations of Quantum Group

Wang, Dingguo; Qingzhong, Ji; Yang, Shilin

doi:10.1081/agb-120003465

Cited by 12 publications

(25 citation statements)

References 9 publications

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“…Note that in general not all finite-dimensional modules over our algebra A are semisimple. The corresponding example is constructed in [6] for the algebra from Section 4.3. A necessary and sufficient condition for all finite-dimensional modules over an ambiskew polynomial ring to be semisimple was given in [8], Theorem 5.1.…”

Section: Casimir Operators and Semisimplicitymentioning

confidence: 99%

“…In this section we will extend the Hopf structure on R to A. We make the following ansatz, guided by [4,6]:…”

Section: The Hopf Structurementioning

confidence: 99%

“…The quantum group U q (sl 2 (C)) has by definition the structure of a Hopf algebra. In [6], an extension of this quantum group to an associative algebra denoted by U q ( f (H, K )) (where f is a Laurent polynomial in two variables) is defined and finite-dimensional representations are studied. The authors show that under certain conditions on f , a Hopf algebra structure can be introduced.…”

Section: Introductionmentioning

confidence: 99%

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Hopf structures on ambiskew polynomial rings

Hartwig

2008

Journal of Pure and Applied Algebra

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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.

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Section: Casimir Operators and Semisimplicitymentioning

confidence: 99%

“…In this section we will extend the Hopf structure on R to A. We make the following ansatz, guided by [4,6]:…”

Section: The Hopf Structurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hopf structures on ambiskew polynomial rings

Hartwig

2008

Journal of Pure and Applied Algebra

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“…In [13], the authors introduce a new quantum algebra U q (f (H, K)), which generalizes the quantum group U q (sl 2 ). Then they obtained statements about its centre by applying the Harish-Chandra homomorphism.…”

Section: The Centre Of ωSl Q (2)mentioning

confidence: 99%

The isomorphisms and the center of weak quantum algebras $\omega sl_q(2) $

Wang¹,

Yang²

2005

Tamkang J. Math.

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“…Several generalizations (or deformations) of the quantized enveloping algebra U q (sl 2 ) have been extensively studied in [2,7,9,17,8]. Especially in [8], a general class of algebras U q ( f (K )) (similar to U q (sl 2 )) was introduced, and their finite-dimensional representations were studied.…”

Section: Introductionmentioning

confidence: 99%

ON REPRESENTATIONS OF QUANTUM GROUPS U_q(f_m(K,H))

Tang

2008

Bull. Austral. Math. Soc.

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We construct families of irreducible representations for a class of quantum groups U q ( f m (K , H ). First, we realize these quantum groups as hyperbolic algebras. Such a realization yields natural families of irreducible weight representations for U q ( f m (K , H )). Second, we study the relationship between U q ( f m (K , H )) and U q ( f m (K )). As a result, any finite-dimensional weight representation of U q ( f m (K , H )) is proved to be completely reducible. Finally, we study the Whittaker model for the center of U q ( f m (K , H )), and a classification of all irreducible Whittaker representations of U q ( f m (K , H )) is obtained.2000 Mathematics subject classification: 17B10, 17B35, 17B37.

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Finite-Dimensional Representations of Quantum Group

Cited by 12 publications

References 9 publications

Hopf structures on ambiskew polynomial rings

Hopf structures on ambiskew polynomial rings

The isomorphisms and the center of weak quantum algebras $\omega sl_q(2) $

ON REPRESENTATIONS OF QUANTUM GROUPS U_q(f_m(K,H))

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Finite-Dimensional Representations of Quantum Group

Cited by 12 publications

References 9 publications

Hopf structures on ambiskew polynomial rings

Hopf structures on ambiskew polynomial rings

The isomorphisms and the center of weak quantum algebras $\omega sl_q(2) $

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

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ON REPRESENTATIONS OF QUANTUM GROUPS U_q(f_m(K,H))