Restricted representations of the Witt superalgebras (Q994277)

The authors study the restricted representations of the simple Witt superalgebra \(\mathfrak{g} = W(n)\) over a field of characteristic \(p > 0\). Recall that \(W(n)\) is the Lie superalgebra of superderivations of the Grassmann superalgebra of rank \(n\). More precisely, the authors adapt the arguments made in the complex case by \textit{V. Serganova} [Lie groups and invariant theory. Providence, RI: American Mathematical Society (AMS), Transl. Ser. 2. Am. Math. Soc. 213, Adv. Math. Sci. 56, 223--239 (2005; Zbl 1106.17006)] to the category \((\mathfrak{T},u(\mathfrak{g}))\)-mod. This is the category of restricted representations of \(\mathfrak{g}\) which are compatible with the action of a suitable algebraic torus \(\mathfrak{T}\). Similar categories have been considered with great success in other contexts, for example \(G_rT\)-modules for modular representations of algebraic groups, see \textit{J. C. Jantzen} [Representations of algebraic groups. 2nd ed. Mathematical Surveys and Monographs 107. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1034.20041)]. The main results are the following. First, the authors obtain a classification and character formulas for simple objects in \((\mathfrak{T},u(\mathfrak{g}))\)-mod. After a subsequent study of indecomposable projectives in that category, they determine explicitly the Cartan invariants of the category \(u(\mathfrak{g})\)-mod. This extends the corresponding result obtained for the classical Witt algebras in [\textit{D. K. Nakano}, Projective modules over Lie algebras of Cartan type. Mem. Am. Math. Soc. 98, No. 470, 84 p. (1992; Zbl 0757.17015)].