Crystal bases for \(U_q(osp (1,2r))\)
The authors construct \(\mathbb{Z}_2\)-graded crystal bases for the quantized universal enveloping algebra \(U\) of the Lie superalgebra \(osp(1,2r)\). It is shown that, as in the Lie algebra case, these crystal bases carry a remarkable combinatorial structure. This paper is arranged as follows. In Section 2, the authors first recall the definition of \(U\) and collect some basic facts about its representations; then they define the concept of a crystal base for \(U\) and consider the crystal bases for the simple modules of \(U\) in the case \(r=1\) as an example. In Section 3, two simple \(U\)-modules and their crystal bases are constructed. In Section 4, the behavior of the crystal bases is considered with respect to taking tensor products of \(U\)-modules. It is shown that these crystal bases possess remarkable combinatorial properties, as in the Lie algebra case. In Section 5, the authors first consider the polarization problem, then they apply the result to complete the proof of the existence of the crystal base for any finite-dimensional \(U\)-module. Finally, two examples are used to illustrate how to use the results to decompose tensor products of \(U\)-modules.
scientific article; zbMATH DE number 1251407