9 research outputs found
R-Matrix and Baxter Q-Operators for the Noncompact SL(N,C) Invariant Spin Chain
The problem of constructing the invariant solutions to the
Yang-Baxter equation is considered. The solutions (-operators) for
arbitrarily principal series representations of are obtained
in an explicit form. We construct the commutative family of the operators
which can be identified with the Baxter operators for the
noncompact spin magnet.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation
We start from known solutions of the Yang-Baxter equation with a spectral
parameter defined on the tensor product of two infinite-dimensional principal
series representations of the group or Faddeev's
modular double. Then we describe its restriction to an irreducible
finite-dimensional representation in one or both spaces. In this way we obtain
very simple explicit formulas embracing rational and trigonometric
finite-dimensional solutions of the Yang-Baxter equation. Finally, we construct
these finite-dimensional solutions by means of the fusion procedure and find a
nice agreement between two approaches
On Complex Gamma-Function Integrals
It was observed recently that relations between matrix elements of certain operators in the spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with symmetry group and as a local Hilbert space give rise to a new type of -function integrals. In this work we present a direct calculation of two such integrals. We also analyse properties of these integrals and show that they comprise the star-triangle relations recently discussed in the literature. It is also shown that in the quasi-classical limit these integral identities are reduced to the duality relations for Dotsenko-Fateev integrals
SL(2,C) Gustafson Integrals
It was shown recently that many of the Gustafson integrals appear in studies of the SL(2, R) spin chain models. One can hope to obtain a generalization of the Gustafson integrals considering spin chain models with a different symmetry group. In this paper we analyse the spin magnet with the SL(2, C) symmetry group in case of open and periodic boundary conditions and derive several new integrals