Tensor representations of $\mathfrak q(\infty)$

Abstract

We introduce a symmetric monoidal category of modules over the direct limit queer superalgebra q(∞). The category can be defined in two equivalent ways with the aid of the large annihilator condition. Tensor products of copies of the natural and the conatural representations are injective objects in this category. We obtain the socle filtrations and formulas for the tensor products of the indecoposable injectives. In addition, it is proven that the category is Koszul self-dual.

On the geometry and representation theory of isomeric matrices

“…(e) follows from [GS,Proposition 4.11] (this only gives injectivity in Rep alg,f (q), but it is easy to see that the representation remains injective in Rep alg (q)). (f) follows from [GS,Lemma 5.13].…”

“…We let Rep alg (q) be the category of algebraic representations, and Rep alg,f (q) the subcategory spanned by objects of finite length. This category is studied in detail in [GS], where it is denoted Trep(g); see [GS,Definition 3.2] for the precise definition. (Similar categories for classical Lie algebras were studied in [DPS,PSe,PSt,SS1].)…”

“…Since T n,m belongs to Trep(g) (see the paragraph following [GS, Definition 3.2]) and Trep(g) is an abelian subcategory of the category of all q-modules, it follows that Rep alg,f (q) ⊂ Trep(g). By [GS,Lemma 3.10] and [GS,Corollary 4.3], every simple object of Trep(g) embeds into some T n,m . By [GS,Proposition 4.10], T n,m is injective in Trep(g), and so we see that every object of Trep(g) embeds into a sum of T n,m 's.…”

“…By [GS,Lemma 3.10] and [GS,Corollary 4.3], every simple object of Trep(g) embeds into some T n,m . By [GS,Proposition 4.10], T n,m is injective in Trep(g), and so we see that every object of Trep(g) embeds into a sum of T n,m 's. This gives the reverse inclusion.…”

Stable representation theory: beyond the classical groups

“…• The papers [DPS,GS,PSe,PSt,Se,SS3] study the stable representation theory of classical (super)groups. The results summarized in §1.2 are all analogs of results from these papers (especially [SS3]).…”