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Journal of Mathematical Physics / Volume 27 / Issue 9

J. Math. Phys. 27, 2217 (1986); http://dx.doi.org/10.1063/1.526993 (11 pages)

Nonlinear equations with superposition formulas and the exceptional group G2. I. Complex and real forms of g2 and their maximal subalgebras

J. Beckers1, V. Hussin1, and P. Winternitz2

1Physique théorique et mathématique, Université de Liège, Institut de Physique au Sart Tilman, B.5. B‐4000, Liège 1, Belgium
2Centre de recherches mathématiques, Université de Montréal, C.P. 6128, Succursale A, Montréal, Québec H3C 3J7, Canada

(Received 12 March 1986; accepted 7 May 1986)

In order to study nonlinear ordinary differential equations with superposition principles, related to the exceptional simple Lie group G2, the complex and real forms of its Lie algebra are examined and their maximal subalgebras are summarized. In particular the parabolic subalgebras of the noncompact real form gNC2(R) are determined. Explicit matrix realizations of the fundamental representation D(1,0) are used and studied in connection with invariant subspaces in a seven‐dimensional (complex or real) vector space. The results are collected in three tables of specific interest for the study of nonlinear differential equations, which will be developed in Paper II of this series.

KEYWORDS and PACS

Keywords

NONLINEAR PROBLEMS, SYMMETRY, DIFFERENTIAL EQUATIONS, LIE GROUPS, ALGEBRA, MATRIX ALGEBRA, IRREDUCIBLE REPRESENTATIONS

PACS

  • 02.20.Sv

    Lie algebras of Lie groups

  • 02.30.Hq

    Ordinary differential equations

  • 11.10.Lm

    Nonlinear or nonlocal theories and models

  • 11.30.Na

    Nonlinear and dynamical symmetries (spectrum-generating symmetries)

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.526993

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

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