PreprintVertex operators and principal subspaces of level one for \(U_q (\widehat{\mathfrak{sl}}_2)\)Slaven KožićAbstractWe consider two different methods of associating vertex algebraic structures with the level \(1\) principal subspaces for \(U_q (\widehat{\mathfrak{sl}}_2)\). In the first approach, we introduce certain commutative operators and study the corresponding vertex algebra and its module. We find combinatorial bases for these objects and show that they coincide with the principal subspace bases found by B. L. Feigin and A. V. Stoyanovsky. In the second approach, we introduce the, socalled nonlocal qvertex algebras, investigate their properties and construct the nonlocal qvertex algebra and its module, generated by FrenkelJing operator and Koyama's operator respectively. By finding the combinatorial bases of their suitably defined subspaces, we establish a connection with the RogersRamanujan identities. Finally, we discuss further applications to quantum quasiparticle relations. Keywords: affine Lie algebra, quantum affine algebra, quantum vertex algebra, principal subspace, quasiparticle, combinatorial basis.AMS Subject Classification: Primary 17B37; secondary (Primary), 17B69 (secondary).
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