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 $Γ\subset U_q'(\mathfrak{so}_n)$ generated by these elements and the corresponding subalgebra $Γ_1\subset U(\mathfrak{so}_n)$. The images of $Γ$ and $Γ_1$ under their respective embeddings into skew group algebras are equal to invariant algebras under certain group actions. We use these facts to show $Γ$ is a Harish-Chandra subalgebra of $U_q'(\mathfrak{so}_n)$ and $Γ_1$ is a Harish-Chandra subalgebra of $U(\mathfrak{so}_n)$.
We study the structure of the Gelfand-Tsetlin subalgebras of a family of idempotent truncations of the generalized blob algebras. We obtain optimal presentations in terms of generators and relations, monomial basis and dimension formula.
Representations of Super Yangians with Gelfand-Tsetlin bases
Shenzhen International Center for Mathematics, Southern University of Science and Technology（深圳数学中心，南方科技大学）；School of Artificial Intelligence, Jianghan University（江汉大学人工智能学院）；School of Mathematics and Statistics, Central China Normal University（华中师范大学数学与统计学院）
研究超杨代数的表示，通过评价同态将其协变张量模视为超杨代数模，并给出简单商模块的 tame 条件，扩展了纳扎罗夫和塔拉索夫的工作。
We construct generic Gelfand-Tsetlin representations of the $\imath$quantum groups $U_q^{\text{tw}}(\mathfrak{so}_3)$ and $U_q^{\text{tw}}(\mathfrak{so}_4)$. These representations are infinite-dimensional analogs to the finite-dimensional irreducible representations provided by Gavrilik and Klimyk. They are quantum analogs of generic Gelfand-Tsetlin representations constructed by Mazorchuk. We give sufficient conditions for irreducibility and provide an upper bound for the length with the help of Casimir elements found by Molev, Ragoucy, and Sorba.
S. Ovsienko证明了$\mathfrak{gl}_n$的Gelfand-Tsetlin簇是等维的（即，它的所有不可约分量具有相同的维数）的维数$\frac{n(n-1)}{2}$。这个结果被称为Ovsienko定理，它在代数的表示论中有重要的意义。本文研究了限制Yangian型$Y_p(\mathfrak{gl}_n)$的Gelfand-Tsetlin簇和$Y_p(\mathfrak{gl}_3)$的等维性.
S. Ovsienko证明了$\mathfrak{gl}_n$的Gelfand-Tsetlin簇是等维的（即它的所有不可约分量都有相同的维数），维数等于$\frac{n(n-1)}{2}$。这一结果在代数的表示论中有重要的意义，特别是Kostant-Wallach映射的零纤维的等维性。在本文中，我们将提出这一结果的推广，并将解决一个弱版本的Ovsienko定理，其中包括正规的情况。
In the paper a realization of representation of a Lie algebra $\mathfrak{sp}_4$ in the space of function on the Lie group $Sp_4$ is considered. We find a function corresponding to a Gelfand-Tsetlin type vector for $\mathfrak{sp}_4$ constructed by D.P. Zhelobenko. This function is expressed though a $A$-hypergeometric function. After developing some new technique we derive analytically formulas for the action of generators of algebra in this base (the were not known before). These formula turn out to be much more difficult than the formulas for the action of generators in the Gelfand-Tsetlin ty…
In their study of the equivariant K-theory of the generalized flag varieties $G/P$, where $G$ is a complex semisimple Lie group, and $P$ is a parabolic subgroup of $G$, Lenart and Postnikov introduced a combinatorial tool, called the alcove paths model. It provides a model for the highest weight crystals with dominant integral highest weights, generalizing the model by semistandard Young tableaux. In this paper, we prove a simple and explicit formula describing the crystal isomorphism between the alcove paths model and the Gelfand-Tsetlin patterns model for type $A$.
A realization of representations of the Lie algebra $\mathfrak{o}_5$ in the space of functions on a group $Spin_5\simeq Sp_4$ is considered. In a representation we take a Gelfand-Tsetlin type base associated with a restriction $\mathfrak{o}_5\downarrow\mathfrak{o}_3$. Such a base is useful is problems appearing in quantum mechanics. We construct explicitely functions on the group that correspond to base vectors. As in the cases of Lie algebras $\mathfrak{gl}_3$, $\mathfrak{sp}_4$ these functions can be expressed through $A$-hypergeometric functions (this does not hold for algebras of these ser…
