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

Abstract

The infinite configuration space of an integrable vertex model based on $U_q(\widehat{gl}(2|2){)}_1$ 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 $U_q(gl(2|2))$-module $\mathring{V}$ and its dual ${\mathring{V}}^{\ast }$ 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 $U_q(gl(2|2))$-module. Explicit expressions are provided for the relevant R-matrix elements. The formal $q\to 0$ 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 ${\mathring{V}}^{\ast }$ 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 $\mathring{V}\otimes {\mathring{V}}^{\ast }$ is given.

45 pages, 5 figures, to appear in Nucl. Phys. B

References in corpus (3)

• Integrable highest weight modules over affine superalgebras and Appell's function
• Vertex Operators of $U_q[\widehat{gl(N|N)}]$ and Highest Weight Representations of $U_q[\widehat{gl(2|2)}]$
• An integrable $U_q(\widehat{gl}(2|2){)}_1$-Model: Corner Transfer Matrices and Young Skew Diagrams