On crystal bases of the \(q\)-analogue of universal enveloping algebras
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 \textit{G. Lusztig} [J. Algebra 131, 466-475 (1990; Zbl 0698.16007)].
\(q\)-analogue of the universal enveloping algebra
scientific article; zbMATH DE number 24068