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Register a new userSerre-Lusztig relations for $imath$quantum groups III
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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.
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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 $(bf U, bf U^imath)$ 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 automorphisms with closed formulas on quasi-split $imath$quantum groups of symmetric Kac-Moody type associated to restricted Weyl groups. The proofs are carried out in the framework of $imath$Hall algebras and reflection functors, thanks to the $imath$Hall algebra realization of $imath$quantum groups in our previous work. Several quantum binomial identities arising along the way are established.
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.
The goal of this paper is to construct quantum analogues of Chevalley groups inside completions of quantum groups or, more precisely, inside completions of Hall algebras of finitary categories. In particular, we obtain pentagonal and other identities in the quantum Chevalley groups which generalize their classical counterparts and explain Faddeev-Volkov quantum dilogarithmic identities and their recent generalizations due to Keller
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