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Inner Type Isometries on Lie Groups
Author(s) -
Miatello Isabel Dotti,
Miatello Roberto J.
Publication year - 1990
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-41.1.151
Subject(s) - maximal torus , lie group , mathematics , isometry (riemannian geometry) , isometry group , pure mathematics , invariant (physics) , torus , simple lie group , compact group , type (biology) , simply connected space , combinatorics , lie algebra , fundamental representation , mathematical physics , geometry , ecology , biology , weight
Let G be a connected Lie group endowed with a left invariant metric and let U = { x ∈ G : g → xgx −1 is an isometry}. In this note we study conditions on H , a compact subgroup of G , to have H = U 0 , 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 ) = G L , if g ≄ su (2).