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, so-called nonlocal $\underline{\mathsf{q}}$-vertex algebras, investigate their properties and construct the nonlocal $\underline{\mathsf{q}}$-vertex algebra and its module, generated by Frenkel-Jing 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 Rogers-Ramanujan identities. Finally, we discuss further applications to quantum quasi-particle relations.

31 pages, 4 figures, minor changes, final version

References in corpus (18)

• Field Algebras
• Vertex-algebraic structure of the principal subspaces of certain A_1^(1)-modules, I: level one case
• Fermionic characters of arbitrary highest-weight integrable sl_{r+1}-modules
• $H_D$-Quantum Vertex Algebras and Bicharacters
• Principal subspaces for quantum affine algebra $U_q(A_n^{(1)})$
• Vertex-algebraic structure of the principal subspaces of certain A_1^(1)-modules, II: higher level case
• Combinatorial bases of Feigin-Stoyanovsky's type subspaces of higher-level standard $\tilde{\gsl}(\ell+1,\C)$-modules
• Presentations of the principal subspaces of the higher-level standard $\widehat{\mathfrak{sl}(3)}$-modules
• Combinatorial bases of Feigin-Stoyanovsky's type subspaces of level 2 standard modules for $D_4^{(1)}$
• Principal $\widehat{sl}(3)$ subspaces and quantum Toda Hamiltonian
• Vertex-algebraic structure of the principal subspaces of level one modules for the untwisted affine Lie algebras of types A,D,E
• A note on the zeroth products of Frenkel-Jing operators
• Combinatorial bases of Feigin-Stoyanovsky's type subspaces of level 1 standard modules for $\tilde{\mathfrak sl}(\ell+1,\C)$
• Quasi-particle bases of principal subspaces for the affine Lie algebras of types $B_l^{(1)}$ and $C_l^{(1)}$
• Characters of Feigin-Stoyanovsky's type subspaces of level one modules for affine Lie algebras of types $A_{\ell }^{(1)}$ and $D_4^{(1)}$
• Combinatorial bases of principal subspaces for affine Lie algebra of type B_2^(1)
• Vertex-algebraic structure of principal subspaces of standard A_2^{(2)}-modules, I
• Recurrence relations for characters of affine Lie algebra $A_{\ell }^{(1)}$

Cited by in corpus (3)

• Combinatorial bases of principal subspaces of modules for twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$
• Quasi-particle bases of principal subspaces of the affine Lie algebra of type $G_2^{(1)}$
• Higher level vertex operators for $U_q({\widehat{\mathfrak{sl}}}_2)$