Complexity of modules over classical Lie superalgebras

Complexity of simple modules over the Lie superalgebra $\mathfrak{osp}(k|2)$

“…For g " osppk|2q (k ą 2), we only consider the full subcategory, F (as in [3,4]), of all finite-dimensional g-supermodules. Note that these are completely reducible over g0 since g0 is semisimple and hence this aligns with the work done in [2] and [10]. For simplicity, we will from now on use the term "module" with the understanding that the prefix "super" is implicit.…”

“…Lie superalgebras and representations. We will mainly use the same notations and conventions from [10] and [16]. We will work over the complex numbers C throughout this paper.…”

“…In [2] the authors computed this categorical invariant for the simple and the Kac modules over glpm|nq and then used a detecting subsuperalgebra f to interpret their computations geometrically. The same was done for the Lie superalgebras ospp2|2nq, ospp3|2q, Dp2, 1; αq, Gp3q, and F p4q in [10]. We carry these computations over osppk|2q and conclude in Theorem 4.0.2 that if Lpλq is a simple module, we have:…”

“…In [10], the author computed the complexity of the simple and the Kac modules over the orthosymplectic Lie superalgebra ospp2|2nq of type C. These computations were interpreted geometrically as in (1.0.1). It is important to note that both types A and C are type I Lie superalgebras and hence similar results were expected.…”

“…It is important to note that both types A and C are type I Lie superalgebras and hence similar results were expected. However, it was shown in [10] that simple modules satisfy (1.0.1) for the type II Lie superalgebras: ospp3|2q, and the three exceptional ones Dp2, 1; αq, Gp3q, and F p4q.…”