Commutation relations of vertex operators for Uq(sl^(M|N))

Abstract

We consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra Uq( sl(M |N )) by using bosonization. We show that vertex operators give a representation of the graded Zamolodchikov-Faddeev algebra by direct computation. Invertibility relations of type-II vertex operators for N > M are very similar to those of type-I for M > N . IntroductionVertex operators and corner transfer matrices are a useful tool in solvable lattice models [1,2,3]. They can be very effective way of calculating correlation functions. In the thermodynamic limit, a half transfer matrix becomes a type-I vertex operator Φ µV λ (z) of the quantum affine algebra U q (g). A type-I vertex operator is, by definition, an intertwiner of theand V (µ) are highest weight representations and V z denotes the evaluation representation [4]. In this paper we consider commutation relations and invertibility relations of the vertex operators for U q ( sl(M |N )) (M = N, M, N ≥ 1) by using bosonization [5]. We show that the vertex operators give a representation of the graded Zamolodchikov-Faddeev algebra by direct computation. Our commutation relations of the vertex operators give a higher-rank generalization of those for U q ( sl(M |1)) [6,7]. A type-. We note that the invertibility relations of the type-II vertex operators for N > M are very similar to those of the type-I for M > N . Our direct computation can be applied to bosonization of vertex operators and a L-operator for the elliptic algebra U q,p ( sl(M |N )) [8,9,10]. Moreover, quantum W -algebra W q,p (sl(M |N )) will arise as fusion of vertex operators for the elliptic algebra [11].The text is organized as follows. In Section 2 we recall bosonization of the quantum affine superalgebra U q ( sl(M |N )) and the vertex operators. In Section 3 we introduce the R-matrix and describe the main theorems. In Section 4 we give a direct proof of the main theorems. In Section 5 we discuss related topics. In Appendix A we summarize normal ordering rules of bosonic operators.where we set δ(z) = m∈Z z m . Here we use the generating functions(2.24)The Chevalley generators are obtained by h i = H i 0 (i = 1, 2, · · · , M + N − 1), (2.25) e i = X +,i 0 , f i = X −,i 0 (i = 1, 2, · · · , M + N − 1), (2.26)

Quadratic relations of the deformed $W$-superalgebra ${\cal W}_{q, t}(\mathfrak{sl}(2|1))$

“…It is interesting to construct the deformed W -algebra W qt (sl(M |N )) by fusion of vertex operators associated with the sl(M |N ) SOS model [24]. Direct proof of commutation relations of vertex operators for the quantum group U q ( sl(M |N )) [38] is very useful for bosonizations of vertex operators associated with the sl(M |N ) SOS model. In [39], the L-operator associated with the elliptic algebra U qp ( sl(N )) was constructed by another fusion of vertex operators associated with the sl(N ) SOS model.…”

Reduced qKZ equation and genuine qKZ equation

“…It is worth to note that the R-operators defined with such a choice of κ V|V (ζ 1 |ζ 2 ) enter commutation relations for quantum vertex operators, see, for example, the paper [37], where the most general supersymmetric case is considered. Now, we transform the graphical equation in figure 22 to the form which can be seen in figure 41.…”

Reduced qKZ equation and genuine qKZ equation

“…) ∞ into equation (4.5), we see that this equation is satisfied if d V|V = 1. It is worth to note that the R-operators defined with such choice a of κ V|V (ζ 1 |ζ 2 ) enter commutation relations for quantum vertex operators, see, for example, the paper [37], where the most general supersymmetric case is considered.…”