Quantum Covering Groups and Quantum Symmetric Pairs

A quantum covering group U is an algebra with parameters q and π such that π squares to 1; it specializes to the usual quantum group at π=1 and to a quantum supergroup of anisotropic type at π=-1. In this dissertation, we establish the Frobenius-Lusztig homomorphism and Lusztig-Steinberg tensor product theorem in the setting of quantum covering groups at roots of 1, recovering Lusztig's constructions for quantum groups at roots of 1 when we specialize at π=1.

We then develop a theory of quantum symmetric pair (U,Ui), where Ui is a coideal subalgebra of U. When specializing at π=1, the pair (U,Ui), reduces to a quantum symmetric pair of G. Letzter and its Kac-Moody generalization by S. Kolb. We give a Serre presentation for Ui of quantum symmetric pairs (U,Ui) for quantum covering groups, introducing the iπ-Serre relations and iπ-divided powers. We also develop a quasi K-matrix in this setting, which leads to a construction of icanonical bases for the highest weight integrable U-modules and their tensor products regarded as Ui-modules, as well as an icanonical basis for the modified form of the iquantum group Ui. Again, specializing at π=1 we recover the Serre presentation of Ui by H. Chen, M. Lu, and W. Wang and the canonical basis construction of H. Bao and W. Wang. The specialization at π=-1 leads to new constructions for quantum supergroups.