Jonas T. Hartwig

Deformations of Lie Algebras using σ-Derivations

In this article we develop an approach to deformations of the Witt and Virasoro algebras based on sigma-derivations. We show that sigma-twisted Jacobi type identity holds for generators of such deformations. For the sigma-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the q-deformations of the Witt and Virasoro algebras associated to q-difference operators, providing also corresponding q-deformed Jacobi identities. (c) 2005 Elsevier Inc. All rights reserved.

Locally finite simple weight modules over twisted generalized Weyl algebras

Abstract We present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. When a certain commutative subalgebra is finitely generated over an algebraically closed field we obtain a classification of a class of locally finite simple weight modules as those induced from simple modules over a subalgebra isomorphic to a tensor product of noncommutative tori. As an application we describe simple weight modules over the quantized Weyl algebra.

Multiparameter twisted Weyl algebras

Abstract We introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordanʼs simple localization of the multiparameter quantized Weyl algebra and Hayashiʼs q -analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs up to isomorphism over a class of twisted generalized Weyl algebras which includes the multiparameter twisted Weyl algebras.

Twisted Generalized Weyl Algebras, Polynomial Cartan Matrices and Serre-Type Relations

A twisted generalized Weyl algebra (TGWA) is defined as the quotient of a certain graded algebra by the maximal graded ideal I with trivial zero component, analogous to how Kac–Moody algebras can be defined. In this article we introduce the class of locally finite TGWAs and show that one can associate to such an algebra a polynomial Cartan matrix (a notion extending the usual generalized Cartan matrices appearing in Kac–Moody algebra theory) and that the corresponding generalized Serre relations hold in the TGWA. We also give an explicit construction of a family of locally finite TGWAs depending on a symmetric generalized Cartan matrix C and some scalars. The polynomial Cartan matrix of an algebra in this family may be regarded as a deformation of the original matrix C and gives rise to quantum Serre relations in the TGWA. We conjecture that these relations generate the graded ideal I for these algebras, and prove it in type A 2.

On the consistency of twisted generalized Weyl algebras

A twisted generalized Weyl algebra A of degree n depends on a base algebra R, n commuting automorphisms s_i of R, n central elements t_i of R and on some additional scalar parameters. In a paper by V.Mazorchuk and L.Turowska (1999) it is claimed that certain consistency conditions for s_i and t_i are sufficient for the algebra to be nontrivial. However, in this paper we give an example which shows that this is false. We also correct the statement by finding a new set of consistency conditions and prove that the old and new conditions together are necessary and sufficient for the base algebra R to map injectively into A. In particular they are sufficient for the algebra A to be nontrivial. We speculate that these consistency relations may play a role in other areas of mathematics, analogous to the role played by the Yang-Baxter equation in the theory of integrable systems.

Hopf structures on ambiskew polynomial rings

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.

The isomorphism problem for multiparameter quantized Weyl algebras

In this note we solve the isomorphism problem for the multiparameter quantized Weyl algebras, in the case when none of the deformation parameters

Extended trigonometric Cherednik algebras and nonstationary Schrödinger equations with delta-potentials

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