#### Vol. 273, No. 1, 2015

 Recent Issues Vol. 299: 1  2 Vol. 298: 1  2 Vol. 297: 1  2 Vol. 296: 1  2 Vol. 295: 1  2 Vol. 294: 1  2 Vol. 293: 1  2 Vol. 292: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Special Issues Submission Guidelines Submission Form Contacts Author Index To Appear ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Other MSP Journals
On Whittaker modules for a Lie algebra arising from the 2-dimensional torus

### Shaobin Tan, Qing Wang and Chengkang Xu

Vol. 273 (2015), No. 1, 147–167
##### Abstract

Let $A$ be the ring of Laurent polynomials in two variables and $B$ be the set of skew derivations of $A$. We denote by $\stackrel{̃}{L}$ the semidirect product of $A$ and $B$, and by $L$ the universal central extension of the derived Lie algebra of $\stackrel{̃}{L}$. We study the Whittaker modules for the Lie algebra $L$. The irreducibilities for the universal Whittaker modules are given. Moreover, a $ℤ$-gradation is defined on the universal Whittaker modules and we determine all $ℤ$-graded irreducible quotients of the reducible universal Whittaker modules.

##### Keywords
Whittaker module, infinite-dimensional Lie algebra, torus
##### Mathematical Subject Classification 2010
Primary: 06B15
Secondary: 17B65, 17B66