Vertex operators and principal subspaces of level one for $U_q (\widehat{\mathfrak{sl}}_2)$
Abstract
We 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 $\underline{\mathsf{q}}$vertex algebras, investigate their properties and construct the nonlocal $\underline{\mathsf{q}}$vertex 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 sum sides of the RogersRamanujan identities. Finally, we discuss further applications to quantum quasiparticle relations.
 Publication:

arXiv eprints
 Pub Date:
 August 2015
 arXiv:
 arXiv:1508.07658
 Bibcode:
 2015arXiv150807658K
 Keywords:

 Mathematics  Quantum Algebra
 EPrint:
 31 pages, 4 figures, minor changes, final version