A Koszul category of representations of finitary Lie algebras

Dan-Cohen, Elizabeth; Penkov, Ivan; Serganova, Vera

doi:10.48550/arxiv.1105.3407

Cited by 7 publications

(18 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let G be the group of all linear operators on V ⊕ W that preserve the pairing (•, •), and that commute with P . Then G is a subgroup of the group of automorphisms of g. Like in the case of gl(∞) (see Theorem 3.4 in [2]) , the large annihilator condition implies that for any γ ∈ G, the twisted module M γ is isomorphic to M. Let W denote the normalizer of h in G. Then for any s ∈ W, if M is a highest weight module with respect to s(b) it is a highest weight module with respect to b. Lemma 3.10. Every simple module in Trep g is isomorphic to V (λ, µ) or ΠV (λ, µ).…”

Section: The Category O ′mentioning

confidence: 98%

See 1 more Smart Citation

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

Grantcharov¹,

Serganova²

2016

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: The Category O ′mentioning

confidence: 98%

“…In [2] it is proven that these categories have enough injective objects and that every object has a finite injective resolution. Furthermore, the algebra of endomorphisms of an injecive cogenerator is described explicitly.…”

Section: Introductionmentioning

confidence: 99%

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

Grantcharov¹,

Serganova²

2016

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…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].) We will require some results from [GS], which we now review.…”

Section: The Generic Categorymentioning

confidence: 99%

On the geometry and representation theory of isomeric matrices

Nagpal,

Sam,

Snowden

2021

Preprint

View full text Add to dashboard Cite

The space of n × m complex matrices can be regarded as an algebraic variety on which the group GL n × GL m acts. There is a rich interaction between geometry and representation theory in this example. In an important paper, de Concini, Eisenbud, and Procesi classified the equivariant ideals in the coordinate ring. More recently, we proved a noetherian result for families of equivariant modules as n and m vary. In this paper, we establish analogs of these results for the space of (n|n) × (m|m) queer matrices with respect to the action of Q n × Q m , where Q n denotes the queer supergroup. Our work is motivated by connections to the Brauer category and the theory of twisted commutative algebras.

show abstract

“…Similarly, we define Rep alg (gl), the category of (wide) algebraic representations, to be the tensor subcategory of Rep(gl) generated by V, V[1], V * , and V * [1]. The category Rep nalg (gl) is equivalent to the category of algebraic representations of gl ∞ studied in [DPS,PSe,PSt,SS2]. It is not semisimple.…”

Section: Polynomial Representations Of Gl Letmentioning

confidence: 99%

“…The orthosymplectic Lie superalgebra osp is the stabilizer of this form inside of gl(W). We define the category Rep alg (osp) of (wide) algebraic representations of osp to be the subcategory of Rep(osp) generated by W. As in the gl case, there is also a narrow category, which is equivalent to the category of algebraic representations of the infinite orthogonal category; this category was studied in [DPS,PSe,PSt,SS2]. Any element of gl = gl(V) acts on W = V ⊕ V * and preserves the form.…”

Section: Polynomial Representations Of Gl Letmentioning

confidence: 99%

Equivariant prime ideals for infinite dimensional supergroups

Laudone¹,

Snowden²

2021

Preprint

View full text Add to dashboard Cite

Let A be a commutative algebra equipped with an action of a group G. The socalled G-primes of A are the equivariant analogs of prime ideals, and of central importance in equivariant commutative algebra. When G is an infinite dimensional group, these ideals can be very subtle: for instance, distinct G-primes can have the same radical. In previous work, the second author showed that if G = GL ∞ and A is a polynomial representation, then these pathologies disappear when G is replaced with the supergroup GL ∞|∞ and A with a corresponding algebra; this leads to a geometric description of G-primes of A. In the present paper, we construct an abstract framework around this result, and apply the framework to prove analogous results for other (super)groups. We give some applications to the queer determinantal ideals.

show abstract

A Koszul category of representations of finitary Lie algebras

Cited by 7 publications

References 0 publications

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

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

On the geometry and representation theory of isomeric matrices

Equivariant prime ideals for infinite dimensional supergroups

Contact Info

Product

Resources

About