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$imath$quantum groups are generalizations of quantum groups which appear as coideal subalgebras of quantum groups in the theory of quantum symmetric pairs. In this paper, we define the notion of classical weight modules over an $imath$quantum group, and study their properties along the lines of the representation theory of weight modules over a quantum group. In several cases, we classify the finite-dimensional irreducible classical weight modules by a highest weight theory.
In the present paper, using the technique of localization, we determine the center of the quantum Schr{o}dinger algebra $S_q$ and classify simple modules with finite-dimensional weight spaces over $S_q$, when $q$ is not a root of unity. It turns out
Let $(bf U, bf U^imath)$ be a quantum symmetric pair of Kac-Moody type. The $imath$quantum groups $bf U^imath$ and the universal $imath$quantum groups $widetilde{bf U}^imath$ can be viewed as a generalization of quantum groups and Drinfeld doubles $w
Let $(bf U, bf U^imath)$ be a quasi-split quantum symmetric pair of arbitrary Kac-Moody type, where quasi-split means the corresponding Satake diagram contains no black node. We give a presentation of the $imath$quantum group $bf U^imath$ with explic
Let $n>1$ be an integer, $alphain{mathbb C}^n$, $bin{mathbb C}$, and $V$ a $mathfrak{gl}_n$-module. We define a class of weight modules $F^alpha_{b}(V)$ over $sl_{n+1}$ using the restriction of modules of tensor fields over the Lie algebra of vector
It is shown that except in three cases conjugacy classes of classical Weyl groups $W(B_{n})$ and $W(D_{n})$ are of type ${rm D}$. This proves that Nichols algebras of irreducible Yetter-Drinfeld modules over the classical Weyl groups $mathbb W_{n}$ (