Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 038, 18 pages      arXiv:1206.6173

On Free Pseudo-Product Fundamental Graded Lie Algebras

Tomoaki Yatsui
Department of Mathematics, Asahikawa Medical University, Asahikawa 078-8510, Japan

Received December 16, 2011, in final form June 14, 2012; Published online June 27, 2012

In this paper we first state the classification of the prolongations of complex free fundamental graded Lie algebras. Next we introduce the notion of free pseudo-product fundamental graded Lie algebras and study the prolongations of complex free pseudo-product fundamental graded Lie algebras. Furthermore we investigate the automorphism group of the prolongation of complex free pseudo-product fundamental graded Lie algebras.

Key words: fundamental graded Lie algebra; prolongation; pseudo-product graded Lie algebra.

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  1. Bourbaki N., Éléments de mathématique. Fasc. XXXVII. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres. Chapitre III: Groupes de Lie, Actualités Scientifiques et Industrielles, No. 1349, Hermann, Paris, 1972.
  2. Bourbaki N., Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968.
  3. Kac V.G., Simple irreducible graded Lie algebras of finite growth, Math. USSR Izv. 2 (1968), 1271-1311.
  4. Kobayashi S., Nagano T., On filtered Lie algebras and geometric structures. I, J. Math. Mech. 13 (1964), 875-907.
  5. Morimoto T., Transitive Lie algebras admitting differential systems, Hokkaido Math. J. 17 (1988), 45-81.
  6. Morimoto T., Tanaka N., The classification of the real primitive infinite Lie algebras, J. Math. Kyoto Univ. 10 (1970), 207-243.
  7. Ochiai T., Geometry associated with semisimple flat homogeneous spaces, Trans. Amer. Math. Soc. 152 (1970), 159-193.
  8. Onishchik A.L., Vinberg È.B., Lie groups and algebraic groups, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990.
  9. Tanaka N., Geometric theory of ordinary differential equations, Report of Grant-in-Aid for Scientific Research MESC Japan, 1989.
  10. Tanaka N., On affine symmetric spaces and the automorphism groups of product manifolds, Hokkaido Math. J. 14 (1985), 277-351.
  11. Tanaka N., On differential systems, graded Lie algebras and pseudogroups, J. Math. Kyoto Univ. 10 (1970), 1-82.
  12. Tanaka N., Projective connections and projective transformations, Nagoya Math. J. 12 (1957), 1-24.
  13. Warhurst B., Tanaka prolongation of free Lie algebras, Geom. Dedicata 130 (2007), 59-69.
  14. Yamaguchi K., Contact geometry of higher order, Japan. J. Math. (N.S.) 8 (1982), 109-176.
  15. Yamaguchi K., Differential systems associated with simple graded Lie algebras, in Progress in Differential Geometry, Adv. Stud. Pure Math., Vol. 22, Math. Soc. Japan, Tokyo, 1993, 413-494.
  16. Yatsui T., On pseudo-product graded Lie algebras, Hokkaido Math. J. 17 (1988), 333-343.

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