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Register a new userSerre-Lusztig relations for $imath$quantum groups
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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.
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The $imath$Serre relations and the corresponding Serre-Lusztig relations are formulated for arbitrary $imath$quantum groups arising from quantum symmetric pairs of Kac-Moody type. This generalizes the main results in [CLW18, CLW20].
let $widetilde{bf U}^imath$ be a quasi-split universal $imath$quantum group associated to a quantum symmetric pair $(widetilde{bf U}, widetilde{bf U}^imath)$ of Kac-Moody type with a diagram involution $tau$. We establish the Serre-Lusztig relations for $widetilde{bf U}^imath$ associated to a simple root $i$ such that $i eq tau i$, complementary to the Serre-Lusztig relations associated to $i=tau i$ which we obtained earlier. A conjecture on braid group symmetries on $widetilde{bf U}^imath$ associated to $i$ disjoint from $tau i$ is formulated.
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.
Expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possibly non-parabolic) reflection subgroups of the Weyl group of type B. We formulate an $imath$Schur duality between an $imath$quantum group of type AIII (allowing black nodes in its Satake diagram) and a Hecke algebra of type B acting on a tensor space, providing a common generalization of Jimbo-Schur duality and Bao-Wangs quasi-split $imath$Schur duality. The quasi-parabolic KL bases on quasi-permutation Hecke modules are shown to match with the $imath$canonical basis on the tensor space. An inversion formula for quasi-parabolic KL polynomials is established via the $imath$Schur duality.
$imath$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.
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