This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. Imaginary Verma Modules and Kashiwara Algebras for U q (  sl(2)). Ben Cox, Vyacheslav Futorny, and Kailash C. Misra Abstract. We consider imaginary Verma modules for quantum affine algebra U q (  sl (2)) and construct Kashiwara type operators and the Kashiwara algebra K q . We show that a certain quotient N − q of U q (  sl (2)) is a simple K q -module. 1. Introduction Corresponding to the standard partition of the root system of an affine Lie algebra into set of positive and negative roots we have a standard Borel subalgebra from which we may induce the standard Verma modules. However, unlike for finite dimensional semisimple Lie algebras for an affine Lie algebra there exists other closed partitions of the root system which are not equivalent to the usual partition of the root system under the Weyl group action. Corresponding to such non- standard partitions we have non-standard Borel subalgebras from which one may induce other non-standard Verma-type modules and these typically contain both finite and infinite dimensional weight spaces. The classification of closed subsets of the root system for affine Kac-Moody algebras was obtained by Jakobsen and Kac [JK85, JK89], and independently by Futorny [Fut90, Fut92]. A categorical setting for these modules was introduced in [Cox94], with certain restrictions, and generalized in [CFM96]. For the algebra  sl(2), the only non-standard modules of Verma-type are the imaginary Verma modules [Fut94]. Drinfeld [Dri85] and Jimbo [Jim85] independently introduced the quantum group U q (g) as q-deformations of universal enveloping algebras of a symmetrizable Kac-Moody Lie algebra g. For generic q, Lusztig [Lus88] showed that integrable highest weight modules of symmetrizable Kac-Moody algebras can be deformed to 2000 Mathematics Subject Classification. Primary 17B37, 17B67; Secondary 81R10, 81B50. Key words and phrases. Quantum affine algebras, Imaginary Verma modules, Kashiwara algebras, simple modules. The authors are grateful to the organizers for the invitation to the conference at Banff where this project was initiated. The first author would like to thank North Carolina State University for the support and hospitality during his numerous visits to Raleigh. The second author was partially supported by Fapesp (processo 2005/60337-2) and CNPq (processo 301743/2007-0). He is grateful to the North Carolina State University for the support and hospitality during his visit to Raleigh. The third author was partially supported by the NSA grant H98230-08-1-0080. 1 105 Contemporary Mathematics Volume 506, 2010 c 20 American Mathematical Society 10