Representations of the Special Lie Superalgebra with p-Character of Height One

F. Duan

Journal of Lie theory, Tome 33 (2023) no. 3, pp. 887-918

Résumé

\def\bk{\mathbf{k}} \def\ggg{{\frak g}} \def\gl{\mathfrak{gl}(n)} \def\uuu{U_{\chi}(\frak g)} Let $\bk$ be an algebraically closed field of prime characteristic and $S(n)$ be the special Lie superalgebra of Cartan type over $\bk$. Define $\bar{S}(n)=S(n)\oplus\bk\mbox{-}\{\xi_1D_1 \}$. So $\bar{S}(n)_0\cong\gl$. Let $\ggg=S(n)$ or $\bar{S}(n)$. We investigate in this paper the representations of $\ggg$ when $\chi$ is restricted or $\mathrm{ht}(\chi)=1$. The main results are listed below.\\ (1) When $\mathrm{ht}(\chi)=1$, the irreducible representations of $U_{\chi}(\ggg)$ are considered. Precisely, the composition factors of the Kac modules are confirmed and the dimensions of simple modules are given.\\ (2) When $\chi=0$ or $\mathrm{ht}(\chi)=1$, the structures of indecomposable projective modules are studied and the Cartan invariants of $\uuu$ are given.\\ (3) When $\chi=0$ or $\mathrm{ht}(\chi)=1$, we show that the representation category over $U_{\chi}(\ggg)$ has only one block (reckoning parities in).

Détail

Classification : 17B10,17B50,17B35
Mots-clés : Special Lie superalgebra, irreducible representations, projective representations, Cartan invariants, block