q-Casimir and q-cut-and-join operators related to Reflection Equation Algebras

Abstract

In this paper we are dealing with the Reflection Equation algebra $\mathcal{M}(R)$, associated with a $G{L}_N$ type Hecke symmetry $R$. In this algebra we define the $q$-analogs of the partial derivatives ${\partial }_j^i$ in generators $m_i^j$ of $\mathcal{M}(R)$. The product $\hat{L}=MD$ of two matrices $M=\Vert m_i^j\Vert$ and $D=\Vert {\partial }_i^j\Vert$ turns out to be a generating matrix of a modified Reflection Equation algebra $\hat{\mathcal{L}}(R)$ which is similar to the universal enveloping algebra $U(g{l}_N)$ in many aspects. Central elements of the modified Reflection Equation algebra give rise to $q$-Casimir operators in a representation of $\hat{\mathcal{L}}(R)$ in the algebra $\mathcal{M}(R)$. We perform a spectral analysis of the first $q$-Casimir operator and formulate a conjecture about the spectrum of the higher ones. At last, we define the normal ordering for the $q$-differential operators and inroduce the $q$-cut-and-join operators. In several explicit examples we express some of $q$-cut-and-join operators via the $q$-Casimir ones by analogy with the classical case.

Some misprints are corrected

References in corpus (2)

• Spectral extension of the quantum group cotangent bundle
• Connection between cut-and-join and Casimir operators