Noncompact sl(N) spin chains: BGG-resolution, Q-operators and alternating sum representation for finite dimensional transfer matrices

TL;DR

This paper investigates the structure of transfer matrices in sl(N) spin chains, demonstrating their factorization into Baxter Q-operators and expressing certain transfer matrices as alternating sums, linking to the Nested Bethe Ansatz.

Contribution

It introduces a novel representation of transfer matrices as alternating sums over infinite-dimensional auxiliary spaces, connecting finite-dimensional cases with Q-operators and the Nested Bethe Ansatz.

Findings

Transfer matrices with infinite-dimensional auxiliary spaces factorize into N commuting Q-operators.

Finite-dimensional transfer matrices can be expressed as alternating sums over infinite-dimensional transfer matrices.

Certain combinations of Q-operators correspond to Q-functions in the Nested Bethe Ansatz.

Abstract

We study properties of transfer matrices in the sl(N) spin chain models. The transfer matrices with an infinite dimensional auxiliary space are factorized into the product of N commuting Baxter Q-operators. We consider the transfer matrices with auxiliary spaces of a special type (including the finite dimensional ones). It is shown that they can be represented as the alternating sum over the transfer matrices with infinite dimensional auxiliary spaces. We show that certain combinations of the Baxter Q-operators can be identified with the Q-functions which appear in the Nested Bethe Ansatz.