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On crystal bases of the $$q$$-analogue of universal enveloping algebras. (English) Zbl 0739.17005
The aim of this paper is to give a proof of the existence and uniqueness theorem of crystal bases for an arbitrary symmetrizable Kac-Moody Lie algebra.
Let $${\mathfrak g}$$ be a symmetrizable Kac-Moody Lie algebra and $$U_ q({\mathfrak g})$$ be the $$q$$-analogue of the universal enveloping algebra $$U({\mathfrak g})$$. For an integrable $$U_ q({\mathfrak g})$$-module $$M$$, the endomorphisms $$\tilde e_ i$$ and $$\tilde f_ i$$ of $$M$$ are introduced. Let $$A$$ be the subring of $$\mathbb{Q}(q)$$ consisting of rational functions regular at $$q=0$$. A pair $$(L,B)$$ is called a (lower) crystal base of $$M$$ if it satisfies the conditions: (1) $$L$$ is a free sub-A-module of $$M$$ such that $$M\simeq\mathbb{Q}(q)\otimes_ AL$$; (2) $$B$$ is a base of the $$\mathbb{Q}$$-vector space $$L/qL$$; (3) $$\tilde e_ iL\subset L$$ and $$\tilde f_ iL\subset L$$ for any $$i$$; (4) $$\tilde e_ iB\subset B\cup\{0\}$$ and $$\tilde f_ iB\subset B\cup\{0\}$$; (5) $$L=\bigoplus_{\lambda\in P}L_ \lambda$$ and $$B=\bigcup_{\lambda\in P}B_ \lambda$$, where $$P$$ is the weight lattice and $$L_ \lambda=L\cap M_ \lambda$$, $$B_ \lambda=B\cap(L_ \lambda/qL_ \lambda)$$; (6) For $$b,b'\in B$$, $$b'=\tilde f_ ib$$ if and only if $$b=\tilde e_ ib'$$. For a dominant integral weight $$\lambda$$, let $$V(\lambda)$$ denote the irreducible $$U_ q({\mathfrak g})$$-module with highest weight $$\lambda$$. Let $$u_ \lambda$$ be the highest weight vector of $$V(\lambda)$$. Let $$L(\lambda)$$ be the smallest sub-$$A$$-module of $$V(\lambda)$$ that contains $$u_ \lambda$$ and that is stable by the actions of $$\tilde f_ i$$. Let $$B(\lambda)$$ be the subset of $$L(\lambda)/qL(\lambda)$$ consisting of the nonzero vectors of the form $$\tilde f_{i_ 1}\dots\tilde f_{i_ k}u_ \lambda\mod qL(\lambda)$$. Then $$(L(\lambda),B(\lambda))$$ is a crystal base of $$V(\lambda)$$ (Theorem 2). Let $$M$$ be an integrable $$U_ q({\mathfrak g})$$- module such that $$M=\bigoplus_{\lambda\in F-Q_ +}M_ \lambda$$ for a finite subset $$F$$ of $$P$$, where $$Q_ +=\oplus\mathbb{N}\alpha_ i$$, and let $$(L,B)$$ be a crystal base of $$M$$. Then there exists an isomorphism $$M\simeq\bigoplus_ j V(\lambda_ j)$$ by which $$(L,B)$$ is isomorphic to $$\bigoplus_ j (L(\lambda_ j),B(\lambda_ j))$$ (Theorem 3). Theorem 2 is proved by the induction on height of weights. The good behavior of crystal bases under the tensor product plays a crucial role in the course of the proof.
In the second part of the paper the author constructs a base named global crystal base of any highest weight irreducible integrable $$U_ q({\mathfrak g})$$-module. In the case of $$A_ n$$, $$D_ n$$ and $$E_ n$$, this coincides with the canonical base introduced by G. Lusztig [J. Algebra 131, 466-475 (1990; Zbl 0698.16007)].
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