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TMF, 2006, Volume 149, Number 1, Pages 3–17 (Mi tmf3823)  

This article is cited in 6 scientific papers (total in 6 papers)

Sylvester–'t Hooft generators and relations between them for $\mathfrak{sl}(n)$ and $\mathfrak{gl}(n|n)$

Ch. Sachse

Max Planck Institute for Mathematics in the Sciences

Abstract: Among the simple finite-dimensional Lie algebras, only $\mathfrak{sl}(n)$ has two finite-order automorphisms that have no common nonzero eigenvector with the eigenvalue one. It turns out that these automorphisms are inner and form a pair of generators that allow generating all of $\mathfrak{sl}(n)$ under bracketing. It seems that Sylvester was the first to mention these generators, but he used them as generators of the associative algebra of all $n\times n$ matrices $\operatorname{Mat}(n)$. These generators appear in the description of elliptic solutions of the classical Yang–Baxter equation, the orthogonal decompositions of Lie algebras, 't Hooft's work on confinement operators in QCD, and various other instances. Here, we give an algorithm that both generates $\mathfrak{sl}(n)$ and explicitly describes a set of defining relations. For simple (up to the center) Lie superalgebras, analogues of Sylvester generators exist only for $\mathfrak{gl}(n|n)$. We also compute the relations for this case.

Keywords: defining relations, Lie algebras, Lie superalgebras

DOI: https://doi.org/10.4213/tmf3823

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English version:
Theoretical and Mathematical Physics, 2006, 149:1, 1299–1311

Bibliographic databases:

Received: 12.12.2005

Citation: Ch. Sachse, “Sylvester–'t Hooft generators and relations between them for $\mathfrak{sl}(n)$ and $\mathfrak{gl}(n|n)$”, TMF, 149:1 (2006), 3–17; Theoret. and Math. Phys., 149:1 (2006), 1299–1311

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. A. V. Lebedev, “On the Bott–Borel–Weil Theorem”, Math. Notes, 81:3 (2007), 417–421  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Lebedev A., “Analogs of the orthogonal, Hamiltonian, Poisson, and contact Lie superalgebras in characteristic 2”, J. Nonlinear Math. Phys., 17, Suppl. 1 (2010), 217–251  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Bouarroudj S., Grozman P., Lebedev A., Leites D., “Divided power (co)homology. Presentations of simple finite dimensional modular Lie superalgebras with Cartan matrix”, Homology, Homotopy Appl., 12:1 (2010), 237–278  crossref  mathscinet  zmath  isi  elib  scopus
    4. Albert V.V., “Quantum Rabi Model for N-State Atoms”, Phys. Rev. Lett., 108:18 (2012), 180401  crossref  adsnasa  isi  scopus
    5. Moroz A., “Quantum Models With Spectrum Generated By the Flows of Polynomial Zeros”, J. Phys. A-Math. Theor., 47:49 (2014), 495204  crossref  mathscinet  zmath  isi  scopus
    6. Albert V.V., Pascazio S., Devoret M.H., “General Phase Spaces: From Discrete Variables to Rotor and Continuum Limits”, J. Phys. A-Math. Theor., 50:50 (2017), 504002  crossref  mathscinet  zmath  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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