Gradings and contractions of affine Kac-Moody algebras

Claudia Daboul; Jamil Daboul; Marc De Montigny

doi:10.1063/1.2940318

Gradings and contractions of affine Kac-Moody algebras

Claudia Daboul, Jamil Daboul, Marc De Montigny

Department of Physics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We study contractions of affine Kac-Moody algebras (KMAs) with respect to their twisted and untwisted Kac-Moody subalgebras and also contract twisted KMA with respect to their affine subalgebras. Our work generalizes the Inönü-Wigner contraction of KMA by Majumdar [J. Math. Phys. 34, 2059 (1993)] in several directions. We compare Inönü-Wigner contractions with respect to twisted subalgebras to another kind of contractions, which we call modulo contractions. We also introduce a novel type of contractions, based on Z -gradings of KMA, which we call level contractions. Along the way we use an algorithm for constructing "Kac-Moody-like algebras" g for any (and not just semisimple) finite-dimensional Lie algebra g.

Original language	English
Article number	063509
Journal	Journal of Mathematical Physics
Volume	49
Issue number	6
DOIs	https://doi.org/10.1063/1.2940318
State	Published - 1 Jan 2008

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1063/1.2940318

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title = "Gradings and contractions of affine Kac-Moody algebras",

abstract = "We study contractions of affine Kac-Moody algebras (KMAs) with respect to their twisted and untwisted Kac-Moody subalgebras and also contract twisted KMA with respect to their affine subalgebras. Our work generalizes the In{\"o}n{\"u}-Wigner contraction of KMA by Majumdar [J. Math. Phys. 34, 2059 (1993)] in several directions. We compare In{\"o}n{\"u}-Wigner contractions with respect to twisted subalgebras to another kind of contractions, which we call modulo contractions. We also introduce a novel type of contractions, based on Z -gradings of KMA, which we call level contractions. Along the way we use an algorithm for constructing {"}Kac-Moody-like algebras{"} g for any (and not just semisimple) finite-dimensional Lie algebra g.",

author = "Claudia Daboul and Jamil Daboul and {De Montigny}, Marc",

note = "Funding Information: J.D. gratefully acknowledges the hospitality of Professor Jean Pierre Gazeau (Paris) and the Laboratoire Astroparticules et Cosmologie (APC), Universit{\'e} Paris 7, where a major part of this manuscript was written. J.D. and M.d.M. thank the Mathematisches Forschungsinstitut Oberwolfach (Germany) where they discussed the manuscript. M.d.M. acknowledges partial financial support from the Natural Sciences and Engineering Research Council of Canada. We thank Professor Daniel Sternheimer for reading the manuscript and for useful comments. ",

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N2 - We study contractions of affine Kac-Moody algebras (KMAs) with respect to their twisted and untwisted Kac-Moody subalgebras and also contract twisted KMA with respect to their affine subalgebras. Our work generalizes the Inönü-Wigner contraction of KMA by Majumdar [J. Math. Phys. 34, 2059 (1993)] in several directions. We compare Inönü-Wigner contractions with respect to twisted subalgebras to another kind of contractions, which we call modulo contractions. We also introduce a novel type of contractions, based on Z -gradings of KMA, which we call level contractions. Along the way we use an algorithm for constructing "Kac-Moody-like algebras" g for any (and not just semisimple) finite-dimensional Lie algebra g.

AB - We study contractions of affine Kac-Moody algebras (KMAs) with respect to their twisted and untwisted Kac-Moody subalgebras and also contract twisted KMA with respect to their affine subalgebras. Our work generalizes the Inönü-Wigner contraction of KMA by Majumdar [J. Math. Phys. 34, 2059 (1993)] in several directions. We compare Inönü-Wigner contractions with respect to twisted subalgebras to another kind of contractions, which we call modulo contractions. We also introduce a novel type of contractions, based on Z -gradings of KMA, which we call level contractions. Along the way we use an algorithm for constructing "Kac-Moody-like algebras" g for any (and not just semisimple) finite-dimensional Lie algebra g.

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Gradings and contractions of affine Kac-Moody algebras

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