2021/10/08 by Dimitri Gurevich, Varvara Petrova, Gurevich, Dimitri +3
Mathematics · Physics and Astronomy · #81R50 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2110.04354
openalex publication_date 2021/10/08 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
In this paper we are dealing with the Reflection Equation algebra cal\nM(R), associated with a GLN type Hecke symmetry R. In this algebra we\ndefine the q-analogs of the partial derivatives \∂ji in generators\nmij of cal M(R). The product L = MD of two matrices\nM=\‖mij\‖ and D=\‖\∂ij\‖ turns out to be a generating matrix of a\nmodified Reflection Equation algebra \ cal L(R) which is similar to the\nuniversal enveloping algebra U(glN) in many aspects. Central elements of the\nmodified Reflection Equation algebra give rise to q-Casimir operators in a\nrepresentation of \ cal L(R) in the algebra cal M(R).\n We perform a spectral analysis of the first q-Casimir operator and\nformulate a conjecture about the spectrum of the higher ones. At last, we\ndefine the normal ordering for the q-differential operators and inroduce the\nq-cut-and-join operators. In several explicit examples we express some of\nq-cut-and-join operators via the q-Casimir ones by analogy with the\nclassical case.\n