Integrable representations of \(U_q(\text{osp}(1,2n))\)
Among the contragredient classical Lie superalgebras, \(G=\text{osp}(1,2n)\) is special since it is the only one whose finite-dimensional representations are completely reducible. Since the representation theory of \(G\) is so close to that of Lie algebras, it is interesting to see if this also holds for the \(q\)-deformation \(U_q(G)\). In the present paper, integrable representations of \(U_q(G)\) are studied. If one works over the field \({\mathbb{C}}(\sqrt{q})\), it is shown that \(U_q(G)\) has integrable simple highest weight representations that are not deformations of representations of \(U(G)\). Furthermore, by considering representations of \(U_q(G)\) which are deformations of \(U(G)\)-modules, the author obtains a PBW theorem for \(U_q(G)\). Finally, the complete reducibility theorem is proved for the category of finite dimensional integrable \(U_q(G)\)-modules.
contragredient classical Lie superalgebras
scientific article; zbMATH DE number 1308207