A $U_q\bigl(\hat{gl}(2|2)\bigr)_1$-Vertex Model: Creation Algebras and Quasi-Particles I

Abstract

The infinite configuration space of an integrable vertex model based on $U_q\bigl(\hat{gl}(2|2)\bigr)_1$ is studied at $q=0$. Allowing four particular boundary conditions, the infinite configurations are mapped onto the semi-standard supertableaux of pairs of infinite border strips. By means of this map, a weight-preserving one-to-one correspondence between the infinite configurations and the normal forms of a pair of creation algebras is established for one boundary condition. A pair of type-II vertex operators associated with an infinite-dimensional $U_q\bigl(gl(2|2)\bigr)$-module $\mathring V$ and its dual $\mathring V^*$ is introduced. Their existence is conjectured relying on a free boson realization. The realization allows to determine the commutation relation satisfied by two vertex operators related to the same $U_q\bigl(gl(2|2)\bigr)$-module. Explicit expressions are provided for the relevant R-matrix elements. The formal $q\to0$ limit of these commutation relations leads to the defining relations of the creation algebras. Based on these findings it is conjectured that the type II vertex operators associated with $\mathring V$ and $\mathring V^*$ give rise to part of the eigenstates of the row-to-row transfer matrix of the model. A partial discussion of the R-matrix elements introduced on $\mathring V\otimes \mathring V^*$ is given.