$$\mathcal {S}$ -交换量子顶点代数和贝特子代数的扬琴变形

Pub Date : 2024-02-02 DOI:10.1007/s00031-023-09837-w

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引用次数: 0

摘要

摘要我们构建了一类新的与归一化杨 R 矩阵相关的量子顶点代数。它们是作为某些 $\mathcal {S}$ -交换量子顶点代数的杨式变形而得到的，它们的 $\mathcal {S}$ -局域性采用了单一的 RTT 关系形式。我们建立了关于它们的表示理论的一些初步结果，然后进一步研究了它们的编织图。特别是，我们证明了它的定点与泊松代数扬琴量子化中的 Bethe 子代数密切相关（\mathcal {O}(\mathfrak {gl}_N((z^{-1}))\)是克雷洛夫和雷布尼科夫最近引入的。最后，我们将换元族的构造扩展到 A 型三角 R 矩阵的情形。

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Yangian Deformations of $$\mathcal {S}$$ -Commutative Quantum Vertex Algebras and Bethe Subalgebras

Abstract

We construct a new class of quantum vertex algebras associated with the normalized Yang R-matrix. They are obtained as Yangian deformations of certain $\mathcal {S}$ -commutative quantum vertex algebras, and their $\mathcal {S}$ -locality takes the form of a single RTT-relation. We establish some preliminary results on their representation theory and then further investigate their braiding map. In particular, we show that its fixed points are closely related with Bethe subalgebras in the Yangian quantization of the Poisson algebra $\mathcal {O}(\mathfrak {gl}_N((z^{-1})))$ , which were recently introduced by Krylov and Rybnikov. Finally, we extend this construction of commutative families to the case of trigonometric R-matrix of type A.

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