\(\imath\)Quantum Covering Groups: Serre presentation and canonical basis

Abstract: In 2016, Bao and Wang developed a general theory of canonical basis for quantum symmetric pairs \((\mathbf{U}, \mathbf{U}^\imath)\), generalizing the canonical basis of Lusztig and Kashiwara for quantum groups and earning them the 2020 Chevalley Prize in Lie Theory. The \(\imath\)divided powers are polynomials in a single generator that generalize Lusztig's divided powers, which are monomials. They can be similarly perceived as canonical basis in rank one, and have closed form expansion formulas, established by Berman and Wang, that were used by Chen, Lu and Wang to give a Serre presentation for coideal subalgebras \(\mathbf{U}^\imath\), featuring novel \(\imath\)Serre relations when \(\tau(i) = i\).

Quantum covering groups, developed by Clark, Hill and Wang, are a generalization that `covers' both the Lusztig quantum group and quantum supergroups of anisotropic type. In this talk, I will talk about how the results for \(\imath\)-divided powers and the Serre presentation can be extended to the quantum covering algebra setting, and subsequently applications to canonical basis for \(\mathbf{U}^\imath_\pi\), the quantum covering analogue of \(\mathbf{U}^\imath\), and quantum covering groups at roots of 1.

combinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

OIST representation theory seminar

Series comments: Timings of this seminar may vary from week to week.