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Register a new userA Drinfeld type presentation of affine $imath$quantum groups II: split BCFG type
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Weinan Zhang
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Recently, Lu and Wang formulated a Drinfeld type presentation for $imath$quantum group $widetilde{{mathbf U}}^imath$ arising from quantum symmetric pairs of split affine ADE type. In this paper, we generalize their results by establishing a current presentation for $widetilde{{mathbf U}}^imath$ of arbitrary split affine type.
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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 explicit $imath$Serre relations. The verification of new $imath$Serre relations is reduced to some new q-binomial identities. Consequently, $bf U^imath$ is shown to admit a bar involution under suitable conditions on the parameters.
$imath$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.
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 $widetilde{bf U}$. In this paper we formulate and establish Serre-Lusztig relations for $imath$quantum groups in terms of $imath$divided powers, which are an $imath$-analog of Lusztigs higher order Serre relations for quantum groups. This has applications to braid group symmetries on $imath$quantum groups.
We establish a three-parameter Schur duality between the affine Hecke algebra of type C and a coideal subalgebra of quantum affine $mathfrak{sl}_n$. At the equal parameter specializations, we obtain Schur dualities of types BCD.
For a Dynkin quiver $Q$ of type ADE and a sum $beta$ of simple roots, we construct a bimodule over the quantum loop algebra and the quiver Hecke algebra of the corresponding type via equivariant K-theory, imitating Ginzburg-Reshetikhin-Vasserots geometric realization of the quantum affine Schur-Weyl duality. Our construction is based on Hernandez-Leclercs isomorphism between a certain graded quiver variety and the space of representations of the quiver $Q$ of dimension vector $beta$. We identify the functor induced from our bimodule with Kang-Kashiwara-Kims generalized quantum affine Schur-Weyl duality functor. As a by-product, we verify a conjecture by Kang-Kashiwara-Kim on the simpleness of some poles of normalized R-matrices for any quiver $Q$ of type ADE.
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