HyperKähler Potentials via Finite-Dimensional Quotients

Piotr Kobak; Andrew Swann

doi:10.1023/A:1013174027792

HyperKähler Potentials via Finite-Dimensional Quotients

^*Corresponding author for this work

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

2 Citations (Scopus)

Abstract

It is known that nilpotent orbits in a complex simple Lie algebra admit hyperKähler metrics with a single function that is a global potential for each of the Kähler structures (a hyperKähler potential). In an earlier paper, the authors showed that nilpotent orbits in classical Lie algebras can be constructed as finite- dimensional hyperKähler quotient of a flat vector space. This paper uses that quotient construction to compute hyperKähler potentials explicitly for orbits of elements with small Jordan blocks. It is seen that the Kähler potentials of Biquard and Gauduchon for SL(n, ℂ)-orbits of elements with X² = 0, are in fact hyperKähler potentials.

Original language	English
Journal	Geometriae Dedicata
Volume	88
Issue	1-3
Pages (from-to)	1-19
Number of pages	19
ISSN	0046-5755
DOIs	https://doi.org/10.1023/A:1013174027792
Publication status	Published - 1 Dec 2001
Externally published	Yes

Keywords

Classical Lie algebras
HyperKähler manifold
HyperKähler potential
HyperKähler quotient
Nilpotent orbit

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10.1023/A:1013174027792

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abstract = "It is known that nilpotent orbits in a complex simple Lie algebra admit hyperK{\"a}hler metrics with a single function that is a global potential for each of the K{\"a}hler structures (a hyperK{\"a}hler potential). In an earlier paper, the authors showed that nilpotent orbits in classical Lie algebras can be constructed as finite- dimensional hyperK{\"a}hler quotient of a flat vector space. This paper uses that quotient construction to compute hyperK{\"a}hler potentials explicitly for orbits of elements with small Jordan blocks. It is seen that the K{\"a}hler potentials of Biquard and Gauduchon for SL(n, ℂ)-orbits of elements with X2 = 0, are in fact hyperK{\"a}hler potentials.",

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AU - Swann, Andrew

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AB - It is known that nilpotent orbits in a complex simple Lie algebra admit hyperKähler metrics with a single function that is a global potential for each of the Kähler structures (a hyperKähler potential). In an earlier paper, the authors showed that nilpotent orbits in classical Lie algebras can be constructed as finite- dimensional hyperKähler quotient of a flat vector space. This paper uses that quotient construction to compute hyperKähler potentials explicitly for orbits of elements with small Jordan blocks. It is seen that the Kähler potentials of Biquard and Gauduchon for SL(n, ℂ)-orbits of elements with X2 = 0, are in fact hyperKähler potentials.

KW - Classical Lie algebras

KW - HyperKähler manifold

KW - HyperKähler potential

KW - HyperKähler quotient

KW - Nilpotent orbit

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JO - Geometriae Dedicata

JF - Geometriae Dedicata

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ER -

HyperKähler Potentials via Finite-Dimensional Quotients

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