Representations of the restricted Cartan type Lie superalgebra \(W(m,n,1)\) (Q409242)

In this paper, the authors study simple modules for the restricted Witt superalgebra \(W(m,n,1)\). They show the sufficient and necessary condition for the simplicity of the Kac module, which extends the results in [\textit{V. Serganova}, Lie groups and invariant theory. Transl. Ser. 2. Am. Math. Soc. 213, Adv. Math. Sci. 56, 223--239 (2005; Zbl 1106.17006)] and [\textit{G. Shen}, Chin. Ann. Math., Ser. B 9, No. 4, 404--417 (1988; Zbl 0689.17011)]. They also prove an important conclusion of the nonrestricted Kac module, that is, when the nonrestricted Kac module is simple, which generalizes analogous results in [\textit{R. R. Holmes}, J. Algebra 237, No. 2, 446--469 (2001; Zbl 1005.17015)] and [\textit{C. Zhang}, J. Algebra 290, No. 2, 408--432 (2005; Zbl 1137.17308)].