Representations of the Lie algebra of derivations for quantum torus

Abstract

Let Der(C q ) be the derivation Lie algebra of the quantum torus C q . We construct a functor from gl d -modules to Der(C q )-modules, which generalizes the constructions obtained by Shen [Scientia Sinica A 29 (1986) 570], Larsson [Internat. J. Modern Phys. A 7 (1992) 6493], and Rao [J. Algebra 182 (1996) 401]. We also give a complete description of the structure of the Der(C q )-modules when all the entries of the quantum torus matrix q = (q ij ) d×d are roots of unity.  2004 Elsevier Inc. All rights reserved.

First cohomology group of rank two Witt algebra to its Larsson modules

“…In 1996, Rao [5] proved that the Der(A d )-module, the image of finite-dimensional irreducible gl d -module under the Larsson functor, is most often irreducible. In 2004, Lin and Tan [3] constructed a functor F α g from gl d -modules V (λ, b) to Der(C q )-modules, where Der(C q ) is the Lie algebra of derivations on quantum torus C q associated with matrix q = (q ij ) d×d . In 2005, Lin and Tan [4] constructed a functor F α g from sl 2 -modules to L q -modules, where L q is the skew derivation Lie algebra over the rank two quantum torus.…”

“…In 2008, Yu [8] constructed a class of irreducible representations for L q with infinite-dimensional weight spaces. In fact, if q = 1, the derivation Lie algebra Der(C q ) is equal to the rank two Witt algebra L , and Der(C q )-modules defined in [3] become L -modules W = F α (V (λ, b)).…”

Indecomposable representations of the Lie algebra of derivations for d-torus

“…Since gl d is a reductive Lie algebra, a finite-dimensional gl d -module is completely reducible if and only if I d is represented by a semisimple endomorphism. The author in [11] (see also [4]) studied the modules arising from irreducible finite-dimensional gl d -modules. But, finite-dimensional gl d -modules are not necessarily completely reducible.…”

Graded Modules Over the q-Analog Virasoro-Like Algebra

“…The quantum torus is one of the main objects in noncommutative geometry, and plays an important role in the classification of extended affine Lie algebras [3]. Meanwhile, the q analog Virasoro like algebra can be regarded as a q deformation of the Virasoro-like algebra introduced and studied by Arnold, de Wit, etc when they try to generalize the Virasoro algebra ( [1,6,11,[13][14][15][16] and [17]). There are some papers devoted to the study of structure and representations of the q analog Virasoro algebra L. C. Jiang and D. Meng studied its derivation Lie algebra and the automorphism group of its derivation Lie algebra [10].…”