Rational R-matrices, centralizer algebras and tensor identities for e6 and e7 exceptional families of Lie algebras

N.J. MacKay; Adele Taylor

doi:10.1063/1.2779960

Rational R-matrices, centralizer algebras and tensor identities for e6 and e7 exceptional families of Lie algebras

N.J. MacKay, Adele Taylor

Mathematics

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

Abstract

We use Cvitanovic's [Group Theory (Princeton University Press, Princeton, NJ, in press) (http://www.nbi.dk/GroupTheory/); Phys. Rev. D 14, 1536 (1976)] diagrammatic techniques to construct the rational solutions of the Yang-Baxter equation [Yang-Baxter Equation in Integrable Systems, edited by M. Jimbo, Advanced Series in Mathematical Physics Vol. 10 (World scientific, Singapore, 1990)] associated with the e(6) and e(7) families of Lie algebras, and thus explain Westbury's [J. Phys. A 36, 2857 (2003)] observations about their uniform spectral decompositions. In doing so, we explore the extensions of the Brauer and symmetric group algebras to the centralizer algebras of e(7) and e(6) on their lowest-dimensional representations and (up to threefold) tensor products thereof, giving bases for them and a range of identities satisfied by the algebras' defining invariant tensors. (C) 2007 American Institute of Physics.

Original language	English
Article number	103507
Pages (from-to)	Art. No. 103507
Number of pages	14
Journal	Journal of Mathematical Physics
Volume	48
Issue number	10
DOIs	https://doi.org/10.1063/1.2779960
Publication status	Published - Oct 2007

Keywords

FACTORIZED S-MATRICES
2 DIMENSIONS
MODELS
REPRESENTATIONS
DIAGRAMS

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10.1063/1.2779960

http://dx.doi.org/10.1063/1.2779960

Cite this

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title = "Rational R-matrices, centralizer algebras and tensor identities for e6 and e7 exceptional families of Lie algebras",

abstract = "We use Cvitanovic's [Group Theory (Princeton University Press, Princeton, NJ, in press) (http://www.nbi.dk/GroupTheory/); Phys. Rev. D 14, 1536 (1976)] diagrammatic techniques to construct the rational solutions of the Yang-Baxter equation [Yang-Baxter Equation in Integrable Systems, edited by M. Jimbo, Advanced Series in Mathematical Physics Vol. 10 (World scientific, Singapore, 1990)] associated with the e(6) and e(7) families of Lie algebras, and thus explain Westbury's [J. Phys. A 36, 2857 (2003)] observations about their uniform spectral decompositions. In doing so, we explore the extensions of the Brauer and symmetric group algebras to the centralizer algebras of e(7) and e(6) on their lowest-dimensional representations and (up to threefold) tensor products thereof, giving bases for them and a range of identities satisfied by the algebras' defining invariant tensors. (C) 2007 American Institute of Physics.",

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T1 - Rational R-matrices, centralizer algebras and tensor identities for e6 and e7 exceptional families of Lie algebras

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N2 - We use Cvitanovic's [Group Theory (Princeton University Press, Princeton, NJ, in press) (http://www.nbi.dk/GroupTheory/); Phys. Rev. D 14, 1536 (1976)] diagrammatic techniques to construct the rational solutions of the Yang-Baxter equation [Yang-Baxter Equation in Integrable Systems, edited by M. Jimbo, Advanced Series in Mathematical Physics Vol. 10 (World scientific, Singapore, 1990)] associated with the e(6) and e(7) families of Lie algebras, and thus explain Westbury's [J. Phys. A 36, 2857 (2003)] observations about their uniform spectral decompositions. In doing so, we explore the extensions of the Brauer and symmetric group algebras to the centralizer algebras of e(7) and e(6) on their lowest-dimensional representations and (up to threefold) tensor products thereof, giving bases for them and a range of identities satisfied by the algebras' defining invariant tensors. (C) 2007 American Institute of Physics.

AB - We use Cvitanovic's [Group Theory (Princeton University Press, Princeton, NJ, in press) (http://www.nbi.dk/GroupTheory/); Phys. Rev. D 14, 1536 (1976)] diagrammatic techniques to construct the rational solutions of the Yang-Baxter equation [Yang-Baxter Equation in Integrable Systems, edited by M. Jimbo, Advanced Series in Mathematical Physics Vol. 10 (World scientific, Singapore, 1990)] associated with the e(6) and e(7) families of Lie algebras, and thus explain Westbury's [J. Phys. A 36, 2857 (2003)] observations about their uniform spectral decompositions. In doing so, we explore the extensions of the Brauer and symmetric group algebras to the centralizer algebras of e(7) and e(6) on their lowest-dimensional representations and (up to threefold) tensor products thereof, giving bases for them and a range of identities satisfied by the algebras' defining invariant tensors. (C) 2007 American Institute of Physics.

KW - FACTORIZED S-MATRICES

KW - 2 DIMENSIONS

KW - MODELS

KW - REPRESENTATIONS

KW - DIAGRAMS

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