Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel

Details Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel

2007-03-29

arxiv.org/abs/math/0703893v2

SIGMA 3 (2007), 060, 31 pages

Mathematics

Quantum Algebra

This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Sym

Scientific paper

10.3842/SIGMA.2007.060

Let $M$ be the tensor product of finite-dimensional polynomial evaluation Yangian $Y(gl_N)$-modules. Consider the universal difference operator $D = \sum_{k=0}^N (-1)^k T_k(u) e^{-k\partial_u}$ whose coefficients $T_k(u): M \to M$ are the XXX transfer matrices associated with $M$. We show that the difference equation $Df = 0$ for an $M$-valued function $f$ has a basis of solutions consisting of quasi-exponentials. We prove the same for the universal differential operator $D = \sum_{k=0}^N (-1)^k S_k(u) \partial_u^{N-k}$ whose coefficients $S_k(u) : M \to M$ are the Gaudin transfer matrices associated with the tensor product $M$ of finite-dimensional polynomial evaluation $gl_N[x]$-modules.

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