Crystal bases for \(U_{q}(\Gamma(\sigma_{1},\sigma_{2},\sigma_{3}))\) (Q2716161)

The author uses an approach similar to that of \textit{G. Benkart, S.-J. Kang} and \textit{M. Kashiwara} [J. Am. Math. Soc. 13, 295-331 (2000; Zbl 0963.17010)] to construct crystal bases for certain modules of the \(q\)-deformation of the universal enveloping algebra of the Lie superalgebra \(\Gamma (\sigma_1, \sigma_2, \sigma_3)\). The construction depends on the crucial fact that the Lie superalgebra \(gl(m,n)\) possesses a natural vector representation \(V\) with a crystal base. For this aim, a simple module \(V\) with a crystal base is constructed first. Then it is shown that certain simple modules possess crystal bases by studying tensor products of \(V\). It turns out that finite-dimensional nontrivial \(\Gamma (\sigma_1, \sigma_2, \sigma_3)\)-modules do not have bases that behave well with respect to tensor products. The modules that are shown to have crystal bases are infinite-dimensional \(\Gamma (\sigma_1, \sigma_2, \sigma_3)\)-modules. NEWLINENEWLINENEWLINEThe main result in this paper is that for any \(\mu= [m,0,n]\) or \([m,n,0]\) (\(m,n\) are nonnegative integers), the simple \(U\)-module \(L(\mu)\) has a polarizable crystal base where \(U= U'\otimes U'\sigma\) and the algebra \(U'\) is the \(\mathbb{Z}_2\)-graded unital associative algebra over \(\mathbb{Q}(q)\) generated by the elements \(E_i, F_i, K_i^\pm\) \((i=1,2,3)\) with certain parities and generating relations.