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–Lie Algebras Canonically Associated to Probability Measures on with All Moments Read chapter Read first chapter

2016 | OriginalPaper | Chapter

The Joseph Ideal for \(\mathfrak {sl}(m|n)\)

Authors : Sigiswald Barbier, Kevin Coulembier

Published in: Lie Theory and Its Applications in Physics

Publisher: Springer Singapore

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Abstract

Using deformation theory, Braverman and Joseph obtained an alternative characterisation of the Joseph ideal for simple Lie algebras, which included even type A. In this note we extend that characterisation to define a remarkable quadratic ideal for \(\mathfrak {sl}(m|n)\). When \(m-n>2\), we prove that the ideal is primitive and can also be characterised similarly to the construction of the Joseph ideal by Garfinkle.

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previous chapter On Finite W-Algebras for Lie Superalgebras in Non-Regular Case
next chapter “Spread” Restricted Young Diagrams from a 2D WZNW Dynamical Quantum Group
Literature
1.
A. Astashkevich, R. Brylinski. Non-local equivariant star product on the minimal nilpotent orbit. Adv. Math. 171 (2002), no. 1, 86–102.
2.
3.
A. Braverman, A. Joseph. The minimal realization from deformation theory. J. Algebra 205 (1998), 113–136.
4.
S.J. Cheng, W. Wang. Dualities and representations of Lie superalgebras. Graduate Studies in Mathematics, 144. American Mathematical Society, Providence, RI, 2012.
5.
K. Coulembier, P. Somberg, V. Souček. Joseph ideals and harmonic analysis for \(\frak osp\it (m|2n)\). Int. Math. Res. Not. IMRN (2014), no. 15, 4291–4340.
6.
M. Eastwood, P. Somberg, V. Souček. Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras. J. Geom. Phys. 57 (2007) 2539–2546.
7.
D. Garfinkle. A new construction of the Joseph ideal. Thesis (Ph.D.) Massachusetts Institute of Technology. 1982.
8.
J. Humphreys. Representations of semisimple Lie algebras in the BGG category  \(\cal{O}\). Graduate Studies in Mathematics, 94. American Mathematical Society, Providence, RI, 2008.
Metadata
Title
The Joseph Ideal for
Authors
Sigiswald Barbier
Kevin Coulembier
Copyright Year
2016
Publisher
Springer Singapore
Book
Lie Theory and Its Applications in Physics

Print ISBN: 978-981-10-2635-5

Electronic ISBN: 978-981-10-2636-2

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-981-10-2636-2_36

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