Almost linear Nash groups

Abstract

A Nash group is said to be almost linear if it has a Nash representation with a finite kernel. Structures and basic properties of these groups are studied.

Keywords

Nash manifold / Nash group / Nash representation / Jordan decomposition / Levi decomposition / Cartan decomposition / Iwasawa decomposition

References

[1] Basu S, Ollack R, Roy M F. Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, 2003, Berlin: Springer-Verlag

[2] Bochnak J, Coste M, Roy M F. Real Algebraic Geometry, 1998, Berlin: Springer-Verlag

[3] Borel A. Linear Algebraic Groups, 1991 2nd ed. New York: Springer-Verlag

[4] Borel A. Semisimple Groups and Riemannian Symmetric Spaces, 1998

[5] Borel A, Mostow G D. On semi-simple automorphisms of Lie algebras. Ann. Math., 1955, 61: 389-504

[6] Cartier P. A primer of Hopf algebras, Frontiers in Number Theory, Physics, and Geometry II, On Conformal Field Theories, Discrete Groups and Renormalization (Papers from the meeting, Les Houches, France, March 9–21, 2003, 537–615), 2007, Berlin: Springer-Verlag

[7] Coste M. An Introduction to Semialgebraic Geometry, RAAG Notes, Institut de Recherche Mathmatiques de Rennes, 2002

[8] Dixmier J. L’application exponentielle dans les groupes de Lie resolubles. Bull. Soc. Math. France, 1957, 85: 113-121

[9] Goodman R, Wallach N R. Symmetry, Representations and Invariants, 2009, New York: Springer-Verlag

[10] Helgason S. Differential Geometry, Lie Groups, and Symmetric Spaces, 2001, Providence, RI: A. M. S.

[11] Jacobson N. Completely reducible Lie algebras of linear transformations. Proc. Amer. Math. Soc., 1951, 2: 105-113

[12] Knapp A W. Lie Groups Beyond an Introduction, 1996 2nd ed. Boston: Birkhäuser

[13] Leptin H, Ludwig J. Unitary Representation Theory of Exponential Lie Groups, 1994, Berlin: W. de Gruyter

[14] Milne J S. Lie Algebras, Algebraic Groups, and Lie Groups, 2012

[15] Mostow G D. Fully reducible subgroups of algebraic groups. Amer. J. Math., 1956, 78: 200-221

[16] Shiota M. Nash Manifolds, 1987, Berlin, Heidelberg, New York, London, Paris, Tokyo: Springer-Verlag

[17] Shiota, M., Nash Functions and Manifolds in Lectures in Real Geometry, F. Broglia (ed.), W. de Gruyter, Berlin, New York, 1996.

[18] Warner F W. Foundations of Differentiable Manifolds and Lie Groups, 1983, New York: Springer-Verlag