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

For any simply-laced GIM Lie algebra ${\mathscr{L}}$ , we present the definition of quantum universal enveloping algebra $U_{q}({\mathscr{L}})$ , and prove that there is a quantum universal enveloping algebra $U_{q}(\mathcal {A})$ of an associated Kac-Moody algebra $\mathcal {A}$ , together with an involution ( $\mathbb {Q}$ -linear) σ, such that $U_{q}({\mathscr{L}})$ is isomorphic to the $\mathbb {Q}(q)$ -extension $\widetilde {S}_{q}$ of the σ-involutory subalgebra Sq of $U_{q}(\mathcal {A})$ . 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 $U_{q}({\mathscr{L}})$ consisting of Lusztig symmetries as a braid group.