The periplectic Brauer algebra III: The Deligne category

Coulembier, Kevin; Ehrig, Michael

doi:10.48550/arxiv.1704.07547

Cited by 5 publications

(8 citation statements)

References 12 publications

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“…The algebra s End(V ⊗r ) is called the signed (or odd) Brauer algebra, and has a diagrammatic basis B, each element describing a string diagram on 2r endpoints, located in 2 rows (r dots in each row). We refer the reader to [BDE + 18] for a detailed description of the basis, and to [CE17b,KT14] for more details on the algebra.…”

Section: Functorialitymentioning

confidence: 99%

“…Periplectic Deligne category. In the periplectic case, a Karoubian additive Deligne category P was constructed in [CE17b,KT14]. This is a Karoubian additive rigid symmetric monoidal category, and is a module category over sVect (hence endowed with an endofunctor Π such that Π 2 ∼ = Id).…”

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confidence: 99%

“…The Karoubian structure of the category P was studied in [CE17b], and its tensor ideals were classified in [Cou17].…”

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confidence: 99%

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Deligne categories and the periplectic Lie superalgebra

Entova-Aizenbud¹,

Serganova²

2018

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We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras p(n) as n → ∞.The paper gives a construction of the tensor category Rep(P ), possessing nice universal properties among tensor categories over the category sVect of finite-dimensional complex vector superspaces.First, it is the "abelian envelope" of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of [EHS15].Secondly, given a tensor category C over sVect, exact tensor functors Rep(P ) → C classify pairs (X, ω) in C where ω : X ⊗ X → Π½ is a non-degenerate symmetric form and X not annihilated by any Schur functor.The category Rep(P ) is constructed in two ways. The first construction is through an explicit limit of the tensor categories Rep(p(n)) (n ≥ 1) under Duflo-Serganova functors. The second construction (inspired by P. Etingof) describes Rep(P ) as the category of representations of a periplectic Lie supergroup in the Deligne category sVect ⊠ Rep(GL t ).An upcoming paper will give results on the abelian and tensor structure of Rep(P ).Contents. Introduction 1 2. Preliminaries and notation 4 3. The infinite periplectic Lie superalgebra 9 4. The subcategories Rep k (p(n)) 12 5. The DS functor 17 6. Construction of the category Rep(P ) 22 7. The first universal property 26 8. The second universal property 29 9. Lower highest weight structure 36 References 39 10. Appendix A: Direct summands of tensor powers 40 11. Appendix B: Affine group schemes in tensor categories 43

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Section: Functorialitymentioning

confidence: 99%

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confidence: 99%

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Deligne categories and the periplectic Lie superalgebra

Entova-Aizenbud¹,

Serganova²

2018

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“…The endomorphism algebras in this supercategory give a diagrammatic realization of Moon's algebra. Since then there has been substantial work applying diagrammatic, combinatorial, and categorical techniques to the study of p(n) and Moon's algebra; see [2,3,4,8,9,10,11,12] and references therein. The present paper further develops this approach to the representation theory of p(n).…”

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confidence: 99%

Webs of Type P

Davidson,

Kujawa,

Muth

2021

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This paper introduces type P web supercategories. They are defined as diagrammatic monoidal -linear supercategories via generators and relations. We study the structure of these categories and provide diagrammatic bases for their morphism spaces. We also prove these supercategories provide combinatorial models for the monoidal supercategory generated by the symmetric powers of the natural module and their duals for the Lie superalgebra of type P.

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“…• This category was introduced in[KT] (under the name "marked Brauer category"), is discussed in [Cou, §2.1] and [Cou2, §2.3] (under the name "periplectic Brauer category"), and also appears in[BE, Example 1.5(iii)] (under the name "odd Brauer supercategory"). • The endomorphism rings in G are the periplectic Brauer algebras introduced by Moon[Mo] (see also[JPW, Proposition 4.1]), and studied in papers of Coulembier-Ehrig[Cou,CE,CE2]. • The usual construction endows G with a triangular structure.…”

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confidence: 99%

The representation theory of Brauer categories I: triangular categories

Sam¹,

Snowden²

2020

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This is the first in a series of papers in which we study representations of the Brauer category and its allies. We define a general notion of triangular category that abstracts key properties of the triangular decomposition of a semisimple complex Lie algebra, and develop a highest weight theory for them. We show that the Brauer category, the partition category, and a number of related diagram categories admit this structure.

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The periplectic Brauer algebra III: The Deligne category

Cited by 5 publications

References 12 publications

Deligne categories and the periplectic Lie superalgebra

Deligne categories and the periplectic Lie superalgebra

Webs of Type P

The representation theory of Brauer categories I: triangular categories

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