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 S_q 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.

对于任何简单刻画的GIM Lie代数\ {{\ mathscr {L}} \\}，我们给出量子通用包络代数\ {U_ {q}（{\ mathscr {L}}）\）的定义，并证明有一个相关的Kac-Moody代数\（\ mathcal {A} \）的量子通用包络代数\（U_ {q}（\ mathcal {A}）\）以及对合（\（\ mathbb {Q } \）- linear）σ，使得\（U_ {q}（{\ mathscr {L}}）\）与\（\ mathbb {Q}（q）\）- extension \（\ widetilde { \（U_ {q}（\ mathcal {A}）\）的σ-不对称子代数S _q的S} _ {q} \）。这一结果给出了一个伯曼的工作（伯曼通讯。代数的量子版本17中只是股价的情况下，3165-3185，1989）。最后，我们将由Lusztig对称性组成的\（U_ {q}（{\ mathscr {L}}）\）自同构群描述为一个编织群。