TY - JOUR

T1 - Duflo–Serganova functor and superdimension formula for the periplectic Lie superalgebra

AU - Entova-Aizenbud, Inna

AU - Serganova, Vera

N1 - Funding Information:
Entova-Aizenbud was supported by the ISF grant 711/18. Serganova was supported by NSF grant 1701532. Part of the work was carried out during the visit of Serganova to Ben Gurion University of the Negev, which was supported by the Faculty of Natural Sciences Distinguished Scientist Visitors Program and by the Center of Advanced Studies in Mathematics in Ben Gurion University.
Publisher Copyright:
© 2022 Mathematical Sciences Publishers.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo–Serganova functor. Given a simple p(n)-module L and a certain odd element x ∈ p(n) of rank 1, we give an explicit description of the composition factors of the p(n−1)-module DSx (L), which is defined as the homology of the complex ∏Mx−→ Mx−→ ∏M, where ∏ denotes the parity-change functor (−) ⊗ ℂ0|1. In particular, we show that this p(n−1)-module is multiplicity-free. We then use this result to give a simple explicit combinatorial formula for the superdimension of a simple integrable finite-dimensional p(n)-module, based on its highest weight. In particular, this reproves the Kac–Wakimoto conjecture for p(n), which was proved earlier by the authors.

AB - In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo–Serganova functor. Given a simple p(n)-module L and a certain odd element x ∈ p(n) of rank 1, we give an explicit description of the composition factors of the p(n−1)-module DSx (L), which is defined as the homology of the complex ∏Mx−→ Mx−→ ∏M, where ∏ denotes the parity-change functor (−) ⊗ ℂ0|1. In particular, we show that this p(n−1)-module is multiplicity-free. We then use this result to give a simple explicit combinatorial formula for the superdimension of a simple integrable finite-dimensional p(n)-module, based on its highest weight. In particular, this reproves the Kac–Wakimoto conjecture for p(n), which was proved earlier by the authors.

KW - Duflo–Serganova functor

KW - Lie superalgebra

KW - periplectic Lie superalgebra

KW - superdimension

UR - http://www.scopus.com/inward/record.url?scp=85134475903&partnerID=8YFLogxK

U2 - 10.2140/ant.2022.16.697

DO - 10.2140/ant.2022.16.697

M3 - Comment/debate

AN - SCOPUS:85134475903

VL - 16

SP - 697

EP - 729

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 3

ER -