On crystal bases of the $Q$-analogue of universal enveloping algebras
On crystal bases of the $Q$-analogue of universal enveloping algebras
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DOI：
10.1215/s0012-7094-91-06321-0
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发表时间：
1991-07
影响因子：
2.5
通讯作者：
M. Kashiwara
中科院分区：
文献类型：
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作者：
M. Kashiwara
0.导论.泛包络代数的q-模拟的概念是由V.G. Drinfeld和M. Jimbo在1985年研究了统计力学中的精确可解模型。这个代数Uq（g）包含一个参数q，并且当q 1时，这与泛包络代数一致。在精确可解模型的上下文中，参数q是温度的参数，并且q 0对应于绝对温度零。因此，我们可以预期q-类似物在q 0处具有简单的结构。在[K1]中，我们将q 0处的研究命名为结晶，并引入了晶基的概念。粗略地说，晶基是在q 0处满足某些公理的Uq（9）-模的基。当g是经典李代数A，，B，，C，和D之一时，证明了U（g）的有限维表示的晶基的存在唯一性. K. Misra和T. Miwa（[M]）证明了U（A1））的基本表示的晶基的存在性，并给出了它的组合描述.本文给出了任意可对称化的Kac-Moody李代数I的晶基存在唯一性定理的证明。此外，我们使这一概念全球化。即借助于一个晶体基，我们构造了一个基，称之为任何最高权不可约可积的整体晶体基
0. Introduction. The notion of the q-analogue of universal enveloping algebras is introduced independently by V. G. Drinfeld and M. Jimbo in 1985 in their study of exactly solvable models in the statistical mechanics. This algebra Uq(g) contains a parameter q, and, when q 1, this coincides with the universal enveloping algebra. In the context of exactly solvable models, the parameter q is that of temperature, and q 0 corresponds to the absolute temperature zero. For that reason, we can expect that the q-analogue has a simple structure at q 0. In [K1] we named crystallization the study at q 0, and we introduced the notion of crystal bases. Roughly speaking, crystal bases are bases of Uq(9)-modules at q 0 that satisfy certain axioms. There, we proved the existence and the uniqueness of crystal bases of finite-dimensional representations of U(g) when g is one of the classical Lie algebras A,, B,, C, and D,. K. Misra and T. Miwa ([M]) proved the existence of a crystal base of the basic representation of U,(A1)) and gave its combinatorial description. The aim of this article is to give the proof of the existence and uniqueness theorem of crystal bases for an arbitrary symmetrizable Kac-Moody Lie algebra I. Moreover, we globalize this notion. Namely, with the aid of a crystal base we construct a base named the global crystal base of any highest weight irreducible integrable