DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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The type-A RTT Yangian in the rational i-normalization

Definition

Fix an integer $N \geq 2$ , the field $C$ , and the ordered vector-space basis $e_{1}, \dots, e_{N}$ of $V_{N} = C^{N}$ . For $1 \leq i, j \leq N$ , write $E_{i j}$ for the matrix unit at zero-based position $(i - 1, j - 1)$ of Matrix units $E_{i j}$ and the Kronecker delta, with $e_{s}$ the column supported at coordinate $s - 1$ ; thus $E_{i j} e_{s} = δ_{j s} e_{i}$ . This reindexes the supplier by $1, \dots, N$ , and ${g l}_{N} (C)$ means $M_{N} (C)$ with the commutator bracket. Tensor products are over $C$ and use the fixed finite tensor convention of The rational spin-1/2 spectral R-matrix; the elementary tensors form the basis supplied by The elementary tensors of two bases form the product basis of the tensor product.

Define the flip $P_{N} \in End (V_{N} \otimes V_{N})$ by $P_{N} (e_{a} \otimes e_{b}) = e_{b} \otimes e_{a}$ , and set $R_{N} (x) = x I + i P_{N}$ , where $i^{2} = - 1$ . For $V_{N}^{\otimes 3}$ , $P_{12}, P_{13}, P_{23}$ swap the indicated positions and $R_{a b} (x) = x I + i P_{a b}$ . Products mean composition, with the rightmost factor applied first.

Let $t_{i j}^{(r)}$ for $1 \leq i, j \leq N$ and $r \geq 1$ be generators, and put $t_{i j} (u) = δ_{i j} + \sum_{r \geq 1} t_{i j}^{(r)} u^{- r}$ and $T_{0} (u) = \sum_{i, j} E_{i j}^{(0)} \otimes t_{i j} (u)$ . On two auxiliary copies of $V_{N}$ , define $T_{1} (u) = \sum_{i, j} E_{i j} \otimes I \otimes t_{i j} (u)$ and $T_{2} (v) = \sum_{i, j} I \otimes E_{i j} \otimes t_{i j} (v)$ , with the third tensor position the coefficient space. The RTT Yangian $Y_{i} ({g l}_{N})$ is the unital associative $C$ -algebra on these generators modulo the coefficientwise formal-series relation

$R_{12} (u - v) T_{1} (u) T_{2} (v) = T_{2} (v) T_{1} (u) R_{12} (u - v) .$

Here the relation means that every coefficient vanishes after expanding each $t_{i j}$ at $u = v = \infty$ .

The normalization is rational and difference-form. At $N = 2$ its $R$ -family is exactly the existing finite supplier The rational spin-1/2 spectral R-matrix; this definition uses that carrier for the rank-two instance and introduces no second $C^{2}$ model.

The proposed coproduct and counit are $Δ (t_{i j} (u)) = \sum_{k} t_{i k} (u) \otimes t_{k j} (u)$ and $ϵ (t_{i j} (u)) = δ_{i j}$ . The formal antipode is $S (T_{0} (u)) = T_{0} (u)^{- 1}$ . Their well-definedness and Hopf identities are proved in The RTT Yangian has its standard Hopf structure ↗.

Depends on

Used by

Dependency tree · two levels

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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)
F. Loebbert, Lectures on Yangian Symmetry (standard reference, not scraped)