Type C blocks of super category $$\mathcal {O}$$

Extension Quiver for Lie Superalgebra q(3)

“…A halfintegral block is a highest weight category both for q(3) and sq (3). The former is also a consequence of general result in [5] for blocks in the category of finite-dimensional representations of q(n) with half-integral weights. The sq(3) standard block is also a highest weight category since it is equivalent to well known A ∞ quiver which also defines the principal block for gl(1|1) [11].…”

“…Algorithms for computing characters of irreducible finite-dimensional representations were obtained in [21,22] using methods of supergeometry and in [3,4] using a categorification approach. Finite-dimensional representations of half-integer weights were studied in detail in [5,6,7]. In [18], the blocks in the category of finite-dimensional q(2)-modules semisimple over the even part were classified and described using quivers and relations.…”

“…Remark 5.5. It was shown in [5] that half-integral blocks in F q(n) of atypicality r are equivalent to modules over the Khovanov are algebra K +∞ r . In our case, r = 1 and the arc algebra is the zig-zag algebra of the semi-infinite linear quiver.…”

On modified extension graphs of a fixed atypicality

“…Let C be a category of representations of a Lie superalgebra g and Irr(C) be the set of isomorphism classes of simple modules in C. Assume that the modules in C are of finite length. 1 In many examples the extension graph of C is bipartite, i.e. there exists a map dex : Irr(C) → Z 2 such that (Dex1) Ext 1 C (L 1 , L 2 ) = 0 if dex(L 1 ) = dex(L 2 ).…”

“…(KM) g = gl(m|n), osp(M|2n) and C = Fin(g); (q; 1 2 ) C = Fin(q m ) 1/2 which is the full subcategory of Fin(q m ) consisting of the modules with "half-integral" weights; (q; C) C is a certain full subcategory of Fin(q m ), see 4.7.…”