156 research outputs found

### Fusion of Baxter's Elliptic $R$-matrix and the Vertex-Face Correspondence

The matrix elements of the $2\times 2$ fusion of Baxter's elliptic
$R$-matrix, $R^{(2,2)}(u)$, are given explicitly. Based on a note by Jimbo, we
give a formula which show that $R^{(2,2)}(u)$ is gauge equivalent to Fateev's
$R$-matrix for the 21-vertex model. Then the crossing symmetry formula for
$R^{(2,2)}(u)$ is derived. We also consider the fusion of the vertex-face
correspondence relation and derive a crossing symmetry relation between the
fusion of the intertwining vectors and their dual vectors.Comment: To appear in the proceedings of the workshop ``Solvable Lattice
Models 2004", July 20--23, 2004, RIMS Koukyuroku, Kyoto Universit

### Dynamical R Matrices of Elliptic Quantum Groups and Connection Matrices for the q-KZ Equations

For any affine Lie algebra ${\mathfrak g}$, we show that any finite
dimensional representation of the universal dynamical $R$ matrix ${\cal
R}(\lambda)$ of the elliptic quantum group ${\cal B}_{q,\lambda}({\mathfrak
g})$ coincides with a corresponding connection matrix for the solutions of the
$q$-KZ equation associated with $U_q({\mathfrak g})$. This provides a general
connection between ${\cal B}_{q,\lambda}({\mathfrak g})$ and the elliptic face
(IRF or SOS) models. In particular, we construct vector representations of
${\cal R}(\lambda)$ for ${\mathfrak g}=A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$,
$D_n^{(1)}$, and show that they coincide with the face weights derived by
Jimbo, Miwa and Okado. We hence confirm the conjecture by Frenkel and
Reshetikhin.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium
on Non-Perturbative and Symmetry Methods in Field Theory (June 2006,
Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA

### Elliptic Quantum Group U_{q,p}(\hat{sl}_2) and Vertex Operators

Introducing an H-Hopf algebroid structure into U_{q,p}(\widedhat{sl}_2), we
investigate the vertex operators of the elliptic quantum group
U_{q,p}(\widedhat{sl}_2) defined as intertwining operators of infinite
dimensional U_{q,p}(\widedhat{sl}_2)-modules. We show that the vertex operators
coincide with the previous results obtained indirectly by using the quasi-Hopf
algebra B_{q,\lambda}(\hat{sl}_2). This shows a consistency of our H-Hopf
algebroid structure even in the case with non-zero central element.Comment: 15 pages. Typos fixed. Version to appear in J.Phys.A :Math.and
Theor., special issue on Recent Developments in Infinite Dimensional Algebras
and Their Applications to Quantum Integrable Systems 200

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