Finite W-superalgebras via super Yangians

We study the Whittaker category N (ζ) of the Lie superalgebra g for an arbitrary character ζ of the even subalgebra of the nilpotent radical associated with a triangular decomposition of g. We prove that the Backelin functor from either the integral subcategory or any strongly typical block of the BGG category to the Whittaker category sends irreducible modules to irreducible modules or zero. The category N (ζ) provides a suitable framework for studying finite W -superalgebras associated with an even principal nilpotent element. For the periplectic Lie superalgebras p(n), we formulate the principal finite W -superalgebras W ζ and establish a Skryabin-type equivalence. For a basic classical and a strange Lie superalgebras, we prove that the category of finite-dimensional modules over a given principal finite W -superalgebra W ζ is equivalent to N (ζ) under the Skryabin equivalence, for a non-singular character ζ. As a consequence, we give a super analogue of Soergel's Struktursatz for a certain Whittaker functor from the integral BGG category O to the category of finite-dimensional modules over W ζ . Contents 1. Introduction 2. Preliminaries 3. Blocks and simple objects in N (ζ) 4. Principal finite W -superalgebras for basic classical Lie superalgebras 5. Principal finite W -superalgebra of periplectic Lie superalgebra p(n) 6. Super Soergel Struktursatz for functor Wh ζ • Γ ζ (−) Appendix A. Whittaker categories of queer Lie superalgebras Appendix B. Equivalence of the categories N (ζ) References MSC 2010: 17B10 17B55

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“…[PS2,Theorem 5.13]). Connections between finite W -superalgebras and super Yangians have been investigated by Peng in [Pe1,Pe2].…”

mentioning

confidence: 99%

Whittaker Categories, Properly Stratified Categories and Fock Space Categorification for Lie Superalgebras

Chen

Cheng

Mazorchuk

2023

Commun. Math. Phys.

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“…Automorphisms. We list the following (anti)automorphisms of Y m|n which are needed in the next section; see [MNO,Proposition 1.12] and [Peng4,(5.22)].…”

Section: Lemma 312 the Following Relations Hold Inmentioning

confidence: 99%

The center of the modular super Yangian $Y_{m|n}$

Chang¹,

Hu²

2022

Preprint

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The present paper is devoted to studying the super Yangian Y m|n associated to the general linear Lie superalgebra gl m|n over a field of positive characteristic. We extend Drinfeld-type presentations of Y m|n and the special super Yangian SY m|n to positive characteristic. Moreover, the center Z(Y m|n ) of Y m|n is described: it is generated by its Harish-Chandra center together with a large p-center. We also study the p-center of SY m|n and provide another description of the p-center of Y m|n in terms of the RTT generators. CONTENTS 1. Introduction 2. The current superalgebra 2.1. The current superalgebra 2.2. Symmetric invariants 2.3. Restricted Lie superalgebra 2.4. The center of U(g) 3. Modular super Yangian Y m|n and Drinfeld-type presentation 3.1. RTT presentation of Y m|n 3.2. Loop filtration 3.3. Gauss decomposition and quasideterminants 3.4. Maps between Super Yangians 3.5. Gauss decomposition of Y m|n 3.6. Drinfeld-type presentation 3.7. Automorphisms 4. The center of Y m|n 4.1. Harish-Chandra center 4.2. Off-diagonal p-central elements 4.3. Diagonal p-central elements 4.4. The center Z(Y m|n ) 5. Modular super Yangian of sl m|n 5.1. The special super Yangian 5.2. The p-centre of SY m|n 5.3. Another description of the p-center of Y m|n References

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“…We list the following (anti)automorphisms of

Y_{m|n}

which are needed in the next section; see [11, Proposition 1.12] and [16, (5.22)]. (1)(‘Multiplication by a power series') For any power series

f(u) \in 1+u^{-1} \mathbb {k}[[u^{-1}]]

, there is an automorphism

\mu _f:Y_{m|n}\rightarrow Y_{m|n}

defined from

\mu _f(T(u))=f(u)T(u)

.…”

Section: Modular Super Yangian Ym|n$y_{m|n}$ and Drinfeld‐type Presen...mentioning

confidence: 99%

The centre of the modular super Yangian Ym|n$Y_{m|n}$

Chang

2022

Journal of London Math Soc

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The present paper is devoted to studying the super Yangian 𝑌 𝑚|𝑛 associated to the general linear Lie superalgebra 𝔤𝔩 𝑚|𝑛 over a field of positive characteristic.We extend Drinfeld-type presentations of 𝑌 𝑚|𝑛 and the special super Yangian 𝑆𝑌 𝑚|𝑛 to positive characteristic.Moreover, the centre 𝑍(𝑌 𝑚|𝑛 ) of 𝑌 𝑚|𝑛 is described: it is generated by its Harish-Chandra centre together with a large 𝑝-centre. We also study the 𝑝-centre of 𝑆𝑌 𝑚|𝑛 and provide another description of the 𝑝-centre of 𝑌 𝑚|𝑛 in terms of the RTT generators.

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Finite W-superalgebras via super Yangians

Cited by 13 publications

References 35 publications

Whittaker Categories, Properly Stratified Categories and Fock Space Categorification for Lie Superalgebras

Whittaker Categories, Properly Stratified Categories and Fock Space Categorification for Lie Superalgebras

The center of the modular super Yangian $Y_{m|n}$

The centre of the modular super Yangian Ym|n$Y_{m|n}$

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