Quantum Symmetric Pairs and Quantum Supergroups at Roots of 1

A quantum group, as conceived by Drinfeld and Jimbo, is the quantization of an enveloping algebra via the quantum parameter v. In analogue with the theory of algebraic groups in prime characteristic, Lusztig laid the foundations of a theory of quantum groups when v has been specialized to a root of 1. Among his fundamental results and constructions are a quantum Frobenius homomorphism, a Steinberg tensor product theorem and the small quantum group.

In this dissertation, we extend the aforementioned results to two related settings. A quantum symmetric pair is the quantization of a symmetric pair of a Lie algebra and its fixed point subalgebra under an involution; the corresponding subalgebra is called an iquantum group. In the first part of the dissertation, we show that Lusztig's Frobenius homomorphism restricts to a map of iquantum groups in finite type. We also formulate the small iquantum group and compute its dimension. A number of elements are shown to be central in the iquantum group at a root of 1. In ADE type, the action of the iquantum group at a root of 1 on the quantized adjoint module gives rise to a Lie algebra isomorphic to the symmetric pair subalgebra.

A quantum covering group is an algebra with parameters v and pi, where pi^2 = 1. When pi is specialized to 1, it is a quantum group of anisotropic Kac-Moody type, and when pi is specialized to -1, it is a quantum supergroup. In the second part, we establish analogues of Lusztig's Frobenius homomorphism and Steinberg tensor product theorem for quantum covering groups. Moreover, we formulate the small quantum covering group; in finite type, we compute its dimension. The specialization of these constructions to pi=1 recovers those of Lusztig.