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ρ (h) = h · g. Notice that, if G is a Lie group, (ii.) · is associative: a ·(b ·c) = (a ·b)·c for
g
ρ g ∈ Diff(G) is a diffeomorphism but not a all a, b, c ∈ R;
homomorphism of the group structure. See also (iii.) + and · satisfy the distributive laws:
left translations and adjoint representations.
a · (b + c) = a · b + a · c and (b +
c) · a = b · a + c · a, for a, b, c ∈ R.
R is called a commutative ring if it is commu-
ring A nonempty set R, with two binary
tative with respect to · .
operations + and · , which satisfy the following
axioms:
rotation An orthogonal tranformation of a
(i.) with respect to +, R is an Abelian group; Euclidean space (V, g). See orthogonal group.

© 2003 by CRC Press LLC
© 2003 by CRC Press LLC