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Journal of the London Mathematical Society 1990 s2-41(1):151-158; doi:10.1112/jlms/s2-41.1.151
© 1990 by London Mathematical Society
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© Oxford University Press

Inner Type Isometries on Lie Groups

Isabel Dotti Miatello and Roberto J. Miatello

Department of Mathematics, Rutgers University New Brunswick, New Jersey 08903, USA

Let G be a connected Lie group endowed with a left invariant metric and let U = {x isin G:g -> xgx–1 is an isometry}. In this note we study conditions on H, a compact subgroup of G, to have H = U0, for some left invariant metric. In the case when H is a torus and G is compact and semisimple we give a necessary and sufficient condition for this to happen. We give several examples and applications to I(G), the full isometry group of G, showing in particular that if G is compact and simple there exist families of metrics such that (1) U and I(G) are highly disconnected, (2) U = {e}, I(G) = GL, if g nsime su(2).

Permanent address: Facultad de Matemática, Astronomia y Física, Universidad Nacional de Córdoba, Valparaíso y R. Martínez, 5032 Córdoba, Argentina


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