Whittaker modules for a Lie algebra of Block type

Abstract

In this paper, we study Whittaker modules for a Lie algebra of Block type. We define Whittaker modules and under some conditions, obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules over this algebra and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra.

Keywords

Whittaker module / Whittaker vector

References

[1] Benkart G., Ondrus M. Whittaker modules for Generalized Weyl algebras. Represent Theory, 2009, 13, 141-164

[2] Christodoulopoulou K. Whittaker modules for Heisenberg algebras and imaginary Whittaker modules for affine Lie algebras. J Algebra, 2008, 320, 2871-2890

[3] Dokovic D., Zhao K. Derivations, isomorphisms and cohomology of generalized Block algebras. Algebra Colloq, 1996, 3, 245-272

[4] Kostant B. On Whittaker vectors and representation theory. Invent Math, 1978, 48, 101-184

[5] Lin W., Tan S. Nonzero level Harish-Chandra modules over the Virasoro-like algebra. J Pure Appl Algebra, 2006, 204, 90-105

[6] Ondrus M. Whittaker modules for U _q(sl₂). J Algebra, 2005, 289, 192-213

[7] Ondrus M., Wiesenr E. Whittaker modules for the Virasoro algebra. J Algebra Appl, 2009, 8, 363-377

[8] Sevostyanov A. Quantum deformation of Whittaker modules and Toda lattice. Duke Math J, 2000, 204, 211-238

[9] Su Y. Quasifinite representations of a Lie algebra of Block type. J Algebra, 2004, 276, 117-128

[10] Su Y. Quasifinite representations of a family of Lie algebras of Block type. J Pure Appl Algebra, 2004, 192, 293-305

[11] Wang B. Whittaker Modules for graded Lie algebras. Algebras and the Representation Theory (to appear); arXiv: 0902.3801

[12] Wu Y., Su Y. Highest weight representations of a Lie algebra of Block type. Sci China, Ser A, 2007, 50, 1267-1279

[13] Xu X. Generalizations of Block algebras. Manuscripta Math, 1999, 100, 489-518

[14] Xu X. Quadratic conformal superalgebras. J Algebra, 2000, 231, 1-38

[15] Yue X, Su Y. Classification of ℤ2-graded modules of the intermediate series over a Lie algebra of Block type. arXiv.math/0611942v2

[16] Zhang H., Zhao K. Representations of the Virasoro-like Lie algebra and its q-analog. Commun Algebra, 1996, 24, 4361-4372

[17] Zhang X., Tan S., Lian H. Whittaker modules for the Schrödinger-Witt algebra. J Math Phys, 2010, 51, 083524

[18] Zhu L., Meng D. Structure of degenerate Block algebras. Algebra Colloq, 2003, 10, 53-62