2022/05/10 by Vasily Krylov, Krylov, Vasily, Leonid Rybnikov +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2205.04700
openalex publication_date 2022/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The loop group G((z-1)) of a simple complex Lie group G has a natural Poisson structure. We introduce a natural family of Poisson commutative subalgebras B(C) ⊂ O(G((z-1)) depending on the parameter C∈ G called classical universal Bethe subalgebras. To every antidominant cocharacter μ of the maximal torus T ⊂ G one can associate the closed Poisson subspace Wμ of G((z-1)) (the Poisson algebra O(Wμ) is the classical limit of so-called shifted Yangian Yμ(\mathfrakg)). We consider the images of B(C) in O(Wμ), that we denote by Bμ(C), that should be considered as classical versions of (not yet defined in general) Bethe subalgebras in shifted Yangians. For regular C centralizing μ, we compute the Poincaré series of these subalgebras. For \mathfrakg=\mathfrakgln, we define the natural quantization Yrtt(\mathfrakgln) of O(Matn((z-1)))) and universal Bethe subalgebras B(C) ⊂ Yrtt(\mathfrakgln). Using the RTT realization of Yμ(\mathfrakgln) (invented by Frassek, Pestun, and Tsymbaliuk), we obtain the natural surjections Yrtt(\mathfrakgln) \twoheadrightarrow Yμ(\mathfrakgln) which quantize the embedding Wμ⊂ Matn((z-1))). Taking the images of B(C) in Yμ(\mathfrakgln) we recover Bethe subalgebras Bμ(C) ⊂ Yμ(\mathfrakgln) proposed by Frassek, Pestun and Tsymbaliuk.