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

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.

💡 Research Summary

The paper investigates the algebraic structure of transfer matrices in non‑compact sl(N) spin chain models, focusing on two complementary perspectives: the factorisation of transfer matrices with infinite‑dimensional auxiliary spaces into Baxter Q‑operators, and the representation of transfer matrices with finite‑dimensional auxiliary spaces as alternating sums of those infinite‑dimensional objects.

First, the authors construct transfer matrices T(u) whose auxiliary space is a generic Verma module (i.e., an infinite‑dimensional highest‑weight representation of sl(N)). By exploiting the underlying Yang‑Baxter algebra, they prove that T(u) can be written as a product of N mutually commuting Baxter Q‑operators,
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