Yang-Baxter equations with two Planck constants

Journal of Physics A: Mathematical and Theoretical. 2016. Vol. 49. No. 1. P. 14003–14021.

Левин А. М., Ольшанецкий М., Zotov A.

We consider Yang–Baxter equations arising from its associative analog and study the corresponding exchange relations. They generate finite-dimensional quantum algebras which have the form of coupled &${\rm{GL}}(N)$; Sklyanin elliptic algebras. Then we proceed to a natural generalization of the Baxter–Belavin quantum R-matrix to the case &${\rm{Mat}}{(N,{\mathbb{C}})}^{\otimes 2}\otimes {\rm{Mat}}{(M,{\mathbb{C}})}^{\otimes 2}.$; It can be viewed as symmetric form of &${\rm{GL}}({NM})\;$; R-matrix in the sense that the Planck constant and the spectral parameter enter (almost) symmetrically. Such type (symmetric) R-matrices are also shown to satisfy the Yang–Baxter like quadratic and cubic equations.

Ключевые слова: integrable systems, quantum R-matrices, Yang-Baxter equation

ПУБЛИКАЦИЯ ПОДГОТОВЛЕНА ПО РЕЗУЛЬТАТАМ ПРОЕКТА: Алгебраическая геометрия и ее приложения: Производные категории; Гомологические и мотивные методы в некоммутативной геометрии; Специальные многообразия; Классическая геометрия; Геометрическая теория представлений; Арифметическая геометрия (2016)