Quantized GIM Algebras and their Images in Quantized Kac-Moody Algebras

For any simply-laced GIM Lie algebra L, we present the definition of quantum universal enveloping algebra U-q (L), and prove that there is a quantum universal enveloping algebra U-q (A) of an associated Kac-Moody algebra A, together with an involution (Q-linear) sigma, such that U-q (L) is isomorphic to the Q(q)-extension (S) over tilde (q) of the sigma-involutory subalgebra S-q of U-q (A). This result gives a quantum version of Berman's work (Berman Comm. Algebra 17, 3165-3185, 1989) in the simply-laced cases. Finally, we describe an automorphism group of U-q (L) consisting of Lusztig symmetries as a braid group.

Automated summary

In this paper, we investigate the quantum universal enveloping algebra U-q(L) of a simply-laced GIM Lie algebra L and establish its isomorphism with U-q(A) of an associated Kac-Moody algebra A. Additionally, we describe the automorphism group of U-q(L) as consisting of Lusztig symmetries forming a braid group.