Generalized symmetric pairs, Lie superalgebras, Dirac operators and Kostant constants
Abstract
In this paper, we present a geometric construction of contragredient Lie superalgebras, with the exceptions of A(m,n) (m,n>0) and G(3). Let g be a complex simple Lie algebra with an order 4 automorphism σ, the eigenspace decomposition being g=g(1)+g(i)+g(−1)+g(−1). When g(i) admits a g(1)-invariant symplectic structure, (g,σ) is called a \emph{generalized symmetric pair with symplectic subspace}. In this paper, we classify all such pairs and construct a natural Lie superalgebra structure on L= L0̅+L1̅ with L0̅=g(1) and L1̅=g(i). Furthermore, we investigate the constants arising from the squares of two distinct Dirac operators: D for g=gσ2+g−σ2, and Ds for L=g(1)+g(i). We prove that the constant C in D2 admits a remarkable factorization C=J*(ΩL0̅)·dim L1̅· h∨sign, where h∨sign is the signed dual Coxeter number of L and J*(ΩL0̅) is a mapped Casimir constant of its even part. This formula reveals an unexpected unification of symmetric space geometry and Lie superalgebra invariants. Finally, we prove that the Kostant constant Ks = .ΩL0̅|L1̅=1/2 is universal for all contragredient Lie superalgebras with non-degenerate Killing form, which implies that the constant Cs in Ds2 satisfies Cs=1/16dim L0̅. We then establish a bijection between such superalgebras and solutions of the superadmissibility index equations EsI I =1/2, EsI = 1/2, recovering the classification from an invariant-theoretic perspective.
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BibTeX
@misc{zhang2026generalized,
title = {{Generalized symmetric pairs, Lie superalgebras, Dirac operators and Kostant constants}},
author = {Mingjing Zhang and Fuhai Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25226},
eprint = {2610.25226},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25226},
}
RIS
TY - UNPB
TI - Generalized symmetric pairs, Lie superalgebras, Dirac operators and Kostant constants
AU - Mingjing Zhang
AU - Fuhai Zhu
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25226
UR - https://arxiv.org/abs/2610.25226
AB - In this paper, we present a geometric construction of contragredient Lie superalgebras, with the exceptions of A(m,n) (m,n>0) and G(3). Let g be a complex simple Lie algebra with an order 4 automorphism σ, the eigenspace decomposition being g=g(1)+g(i)+g(−1)+g(−1). When g(i) admits a g(1)-invariant symplectic structure, (g,σ) is called a \emph{generalized symmetric pair with symplectic subspace}. In this paper, we classify all such pairs and construct a natural Lie superalgebra structure on L= L_0̅+L_1̅ with L_0̅=g(1) and L_1̅=g(i). Furthermore, we investigate the constants arising from the squares of two distinct Dirac operators: D for g=g^σ^2+g^−σ^2, and D_s for L=g(1)+g(i). We prove that the constant C in D^2 admits a remarkable factorization C=J^*(Ω_L_0̅)·dim L_1̅· h^∨_sign, where h^∨_sign is the signed dual Coxeter number of L and J^*(Ω_L_0̅) is a mapped Casimir constant of its even part. This formula reveals an unexpected unification of symmetric space geometry and Lie superalgebra invariants. Finally, we prove that the Kostant constant K_s = .Ω_L_0̅|_L_1̅=1/2 is universal for all contragredient Lie superalgebras with non-degenerate Killing form, which implies that the constant C_s in D_s^2 satisfies C_s=1/16dim L_0̅. We then establish a bijection between such superalgebras and solutions of the superadmissibility index equations E_s^I I =1/2, E_s^I = 1/2, recovering the classification from an invariant-theoretic perspective.
ER -