Research

Papers

1. Canonical bases arising from ıquantum covering groups
   arXiv version
   Abstract: (click to expand)

   For ıquantum covering groups (U,U^ı) of super Kac-Moody type, we construct ı-canonical bases for the highest weight integrable U-modules and their tensor products regarded as U^ı-modules, as well as a canonical basis for the modified form of the ıquantum covering group U^ı, using the ı^π-divided powers, rank one canonical basis for U^ı.




2. A Serre Presentation for the ıQuantum Covering Groups
   arXiv version
   Abstract: (click to expand)

   Let (U,U^ı) be a quasi-split quantum symmetric pair of Kac-Moody type. The ıquantum group U^ı admits a Serre presentation featuring the ı-Serre relations in terms of ı-divided powers. Generalizing this result, we give a Serre presentation U_π^ı of quantum symmetric pairs (U_π,U_π^ı) for quantum covering algebras U_π, which have an additional parameter π that specializes to the Lusztig quantum group when π=1 and quantum supergroups of anisotropic type when π=−1. We give a Serre presentation for U_π^ı, introducing the ı^π-Serre relations and ı^π-divided powers.




3. Quantum Supergroups VI: Roots of 1 (with Thomas Sale and Weiqiang Wang)
   Lett. Math. Phys. 109 (2019), pp. 2753-2777,
   arXiv version   journal version
   Abstract: (click to expand)

   A quantum covering group is an algebra with parameters q and π subject to π²=1, and it admits an integral form; it specializes to the usual quantum group at π=1 and to a quantum supergroup of anisotropic type at π=−1. In this paper we establish the Frobenius–Lusztig homomorphism and Lusztig–Steinberg tensor product theorem in the setting of quantum covering groups at roots of 1. The specialization of these constructions at π=1 recovers Lusztig’s constructions for quantum groups at roots of 1.