2019/08/17 by Feifei Duan, Duan, Fei-Fei, Bin Shu +3 · 1 citation
Mathematics · Physics and Astronomy · #17B10 #17B66 #17B70 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1908.06251
openalex publication_date 2019/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the parabolic BGG categories for graded Lie superalgebras of Cartan type over complex numbers. The gradation of such a Lie superalgebra \ggg naturally arises, with the zero component \ggg0 being a reductive Lie algebra. We first show that there are only two proper parabolic subalgebras containing Levi subalgebra \ggg0: the ``maximal one" \sfpmax and the ``minimal one" \sfpmin. Furthermore, the parabolic BGG category arising from \sfpmax, essentially turns out to be a subcategory of the one arising from \sfpmin. Such a priority of \sfpmin in the sense of representation theory reduces the question to the study of the ``minimal parabolic" BGG category \comi associated with \sfpmin. We prove the existence of projective covers of simple objects in these categories, which enables us to establish a satisfactory block theory. Most notably, our main results are as follows: (1) We classify and obtain a precise description of the blocks of \comi. (2) We investigate indecomposable tilting and indecomposable projective modules in \comi, and compute their character formulas.