Generic Gelfand-Tsetlin Modules of Quantized and Classical Orthogonal Algebras

TL;DR

This paper constructs and analyzes infinite-dimensional modules for quantum and classical orthogonal algebras, embedding them into skew group algebras and identifying Harish-Chandra subalgebras, extending prior finite-dimensional module work.

Contribution

It introduces new infinite-dimensional modules for $U_q'( ext{so}_n)$ and $U( ext{so}_n)$ with rational matrix coefficients and establishes their algebra embeddings and subalgebra properties.

Findings

Constructed infinite-dimensional modules with rational matrix coefficients.

Embedded algebras into skew group algebras of shift operators.

Identified Harish-Chandra subalgebras within quantum and classical orthogonal algebras.

Abstract

We construct infinite-dimensional analogues of finite-dimensional simple modules of the nonstandard $q$-deformed enveloping algebra $U_q'(\mathfrak{so}_n)$ defined by Gavrilik and Klimyk, and we do the same for the classical universal enveloping algebra $U(\mathfrak{so}_n)$. In this paper we only consider the case when $q$ is not a root of unity, and $q\to 1$ for the classical case. Extending work by Mazorchuk on $\mathfrak{so}_n$, we provide rational matrix coefficients for these infinite-dimensional modules of both $U_q'(\mathfrak{so}_n)$ and $U(\mathfrak{so}_n)$. We use these modules with rationalized formulas to embed the respective algebras into skew group algebras of shift operators. Casimir elements of $U_q'(\mathfrak{so}_n)$ were given by Gavrilik and Iorgov, and we consider the commutative subalgebra $\Gamma\subset U_q'(\mathfrak{so}_n)$ generated by these elements and the corresponding subalgebra $\Gamma_1\subset U(\mathfrak{so}_n)$. The images of $\Gamma$ and $\Gamma_1$ under their respective embeddings into skew group algebras are equal to invariant algebras under certain group actions. We use these facts to show $\Gamma$ is a Harish-Chandra subalgebra of $U_q'(\mathfrak{so}_n)$ and $\Gamma_1$ is a Harish-Chandra subalgebra of $U(\mathfrak{so}_n)$.