On a Vector Field Formula for the Lie Algebra of a Homogeneous Space

Kantor, I. L.

doi:10.1006/jabr.2000.8504

Cited by 6 publications

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References 3 publications

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“…It has been shown [52] (see also [54]) that there is an injective homomorphism χ : g → M(g − ) given by…”

Section: 26)mentioning

confidence: 99%

Exceptional Lie algebras and M-theory

Palmkvist

2009

Preprint

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In this thesis we study algebraic structures in M-theory, in particular the exceptional Lie algebras arising in dimensional reduction of its low energy limit, eleven-dimensional supergravity. We focus on e 8 and its infinite-dimensional extensions e 9 and e 10 . We review the dynamical equivalence, up to truncations on both sides, between elevendimensional supergravity and a geodesic sigma model based on the coset E 10 /K(E 10 ), where K(E 10 ) is the maximal compact subgroup. The description of e 10 as a graded Lie algebra is crucial for this equivalence. We study generalized Jordan triple systems, which are closely related to graded Lie algebras, and which may also play a role in the description of M2-branes using three-dimensional superconformal theories. Professor Martin Cederwall and my examiner Professor Bengt E. W. Nilsson. Furthermore, I would like to thank Axel Kleinschmidt and our collaborators in Groningen: Eric A. Bergshoeff, Olaf Hohm and Teake A. Nutma, especially for their efforts to finish one of the papers when the deadline for my thesis was approaching. I am grateful to Jonas Hartwig, Ling Bao and especially Daniel Persson for carefully reading parts of the manuscript and giving me many valuable comments. I am also grateful to Christoffer Petersson for generously lending me his sofa and his office during my visits in Göteborg. Finally, among the students and postdocs at the Albert Einstein Institute I would especially like to thank

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“…It has been shown [52] (see also [54]) that there is an injective homomorphism χ : g → M(g − ) given by…”

Section: 26)mentioning

confidence: 99%

Exceptional Lie algebras and M-theory

Palmkvist

2009

Preprint

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“…For any decomposition of a Lie algebra G into a direct sum G " H 'E of a subalgebra H and a complementary subspace E, there is formula for a nonlinear realisation of G on E given by Kantor [1]. The conformal realisation of a semisimple Lie algebra with a 3-grading G " G ´1 ' G 0 ' G 1 is obtained from this formula in the special case where H " G 0 ' G 1 and E " G ´1.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear realisations of Lie superalgebras

Palmkvist¹

2020

Preprint

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For any decomposition of a Lie superalgebra G into a direct sum G " H ' E of a subalgebra H and a subspace E, we construct a nonlinear realisation of G on E. The result generalises a theorem by Kantor from Lie algebras to Lie superalgebras. When G is a differential graded Lie algebra, it gives a construction of an associated L 8 -algebra.2 The Z-graded Lie algebra associated to a vector spaceWe start with an arbitrary vector space U 1 (over some arbitrary field), from which we define vector spaces U 0 , U ´1, U ´2, . . . recursively by(2.1) for p " 1, 2, . . .. Thus U ´p`1 consists of all linear maps from U 1 to U ´p`2 , and in particular U 0 " End U 1 . Let A p P U ´p`1 , for some p " 1, 2, . . ., and let x 1 , x 2 , . . . P U 1 . Then A p px 1 q P U ´p`2 and if p ě 2, this means that A p px 1 qpx 2 q " `Ap px 1 q ˘px 2 q is an element in U ´p`3 .

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“…Our method is useful for studying nonlinear realizations of Lie algebras [13,14]. Any graded Lie algebra can be realized nonlinearly on its subspaces of negative (or positive) degree [15]. This nonlinear realization can be expressed in terms of the corresponding generalized Jordan triple system.…”

Section: Introductionmentioning

confidence: 99%

“…13,14 Any graded Lie algebra can be realized nonlinearly on its subspaces of negative ͑or positive͒ degree. 15 This nonlinear realization can be expressed in terms of the corresponding generalized Jordan triple system. When this in turn is obtained from an original one for some n in the way that we will describe, then the nonlinear realization can be expressed in terms of the original generalized Jordan triple system.…”

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confidence: 99%

Generalized conformal realizations of Kac–Moody algebras

Palmkvist

2009

Journal of Mathematical Physics

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We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n = 1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of Hermitian 3 ϫ 3 matrices over the division algebras R , C , H , O, the construction gives the exceptional Lie algebras f 4 , e 6 , e 7 , e 8 for n = 2. Moreover, we obtain their infinite-dimensional extensions for n Ն 3. In the case of 2 ϫ 2 matrices, the resulting Lie algebras are of the form so͑p + n , q + n͒ and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature ͑p , q͒.

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On a Vector Field Formula for the Lie Algebra of a Homogeneous Space

Cited by 6 publications

References 3 publications

Exceptional Lie algebras and M-theory

Exceptional Lie algebras and M-theory

Nonlinear realisations of Lie superalgebras

Generalized conformal realizations of Kac–Moody algebras

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