Crystal bases for quantum superalgebras (Q2741028)

In [Commun. Math. Phys. 133, 249-260 (1990; Zbl 0724.17009)], \textit{M. Kashiwara} introduced the concent of a crystal base of an integrable module over a quantized enveloping algebra associated to a symmetrizable Kac-Moody Lie algebra. This is a remarkable basis at \(q=0\), with many excellent properties. It is also closely linked, via the global crystal basis, to the canonical basis of \textit{G. Lusztig} [Introduction to quantum groups (Prog. Math. 110. Birkhäuser, Boston) (1993; Zbl 0788.17010)].NEWLINENEWLINENEWLINEIn this paper, the authors generalise this theory to give a general crystal base theory for quantum superalgebras. As a result, they are able to give interesting explicit descriptions for the orthosymplectic Lie superalgebra \(\text{osp}(1,2n)\), affine Kac-Moody superalgebras and the general linear Lie superalgebra \(\text{gl}(m,n)\) in terms of tableaux, which they compare with work that has already been completed in this direction: \textit{G. Benkart, S.-J. Kang, M. Kashiwara} [J. Am. Math. Soc. 13, 295-331 (2000; Zbl 0963.17010)] (for the general linear case); \textit{I. M. Musson} and \textit{Y.-M. Zou} [J. Algebra 210, 514-534 (1998; Zbl 0918.17009)] and \textit{S.-N. Choi} [Crystal graphs for the orthosymplectic quantum superalgebra \(U_q(osp(1,2n))\), in preparation] (for the orthosymplectic case); and \textit{K. Jeong} [J. Algebra 237, 562-590 (2001; Zbl 1024.17010)] (for the Kac-Moody case).NEWLINENEWLINEFor the entire collection see [Zbl 0963.00024].