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The rational i-normalization is the standard RTT normalization after rescaling

Statement

Let $R_{i} (x) = x I + i P_{N}$ be the normalization of The type-A RTT Yangian in the rational i-normalization, and define the standard polynomial Yang matrix $R_{0} (z) = z I - P_{N}$ . The change of spectral variable $z = i u$ , with $\tilde{T} (z) = T (- i z)$ , converts the RTT relation for $R_{i}$ to the standard RTT relation for $R_{0}$ . Under this change the evaluation parameter $a$ becomes $i a$ .

Facts & Assumptions

Given: $N \geq 2$ , $R_{i} (x) = x I + i P_{N}$ , the RTT relation, and the evaluation-module formula of Evaluation modules of the type-A RTT Yangian.

[F1]

The type-A RTT Yangian in the rational i-normalization: $R_{i} (x) = x I + i P_{N}$ and $R_{i} (u - v) T_{1} (u) T_{2} (v) = T_{2} (v) T_{1} (u) R_{i} (u - v)$ .

[F2]

Evaluation modules of the type-A RTT Yangian: $t_{i j} (u) = δ_{i j} I - i ρ (E_{i j}) / (u - a)$ .

Proof

1.1givenF1algebra

Direct substitution gives $R_{i} (- i (z - w)) = - i ((z - w) I - P_{N}) = - i R_{0} (z - w)$ .

2.1givenF1step 1.1algebra

Substitute $u = - i z$ and $v = - i w$ in the RTT relation [F1]. By step 1.1, each side has the same nonzero scalar factor $- i$ , which cancels; the resulting relation is $R_{0} (z - w) {\tilde{T}}_{1} (z) {\tilde{T}}_{2} (w) = {\tilde{T}}_{2} (w) {\tilde{T}}_{1} (z) R_{0} (z - w)$ .

3.1givenF2algebra∎

In the evaluation module, substitute $u = - i z$ into [F2]. Since $- i z - a = - i (z - i a)$ , one obtains $t_{i j} (- i z) = δ_{i j} I + ρ (E_{i j}) / (z - i a)$ . Thus the shifted standard evaluation parameter is exactly $i a$ .

Depends on

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Sources

A. I. Molev, M. L. Nazarov, and G. I. Olshanski, Yangians and classical Lie algebras (standard reference, not scraped)
A. P. Isaev, Lectures on Quantum Groups and Yang-Baxter Equations (standard reference, not scraped)