Higher level vertex operators for $$U_q \left( \widehat{\mathfrak {sl}}_2\right) $$ U q sl ^ 2

Abstract

Abstract. We study graded nonlocal q-vertex algebras and we prove that they can be generated by certain sets of vertex operators. As an application, we consider the family of graded nonlocal q-vertex algebras V c,1 , c ≥ 1, associated with the principal subspaces W (cΛ 0 ) of the integrable highest weight U q ( sl 2 )-modules L(cΛ 0 ). Using quantum integrability, we derive combinatorial bases for V c,1 and compute the corresponding character formulae. IntroductionIn their work [LP], J. Lepowsky and M. Primc found the, so-called integrability condition for the affine Kac-Moody Lie algebra sl 2 ,on a level c integrable sl 2 -module. In general, (0.1) holds on an arbitrary level c integrable g-module, when the simple root α is replaced by the maximal root of the untwisted affine Kac-Moody Lie algebra g. In this paper, we continue our research on vertex algebraic structures arising from FrenkelJing operators x ± 1 (z) for U q ( sl 2 ), which was initiated in [Ko3]. So far there were several fruitful approaches to associating vertex algebra-like theories with the various quantum objects, such as quantum affine algebras or Yangians, which resulted in some fundamental results and important constructions (cf. [AB],[B],[EK],[FR],[L1]-[L4]). However, motivated by the role of integrability (0.1) in the representation theory of the affine Kac-Moody Lie algebras, we introduce graded nonlocal q-vertex algebras, certain new structures which are designed to make use of quantum integrability (0.2).2000 Mathematics Subject Classification. 17B37 (Primary), 17B69 (Secondary).