By Jacques Faraut.

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Since ad F(t) < log 2 this can be written F (t) = Exp(ad F(t)) Y. 46 Linear Lie groups We can also write F (t) = Ad(exp F(t) Y = Ad(exp X ) Ad(exp tY ) Y = Exp(ad X ) Exp(ad tY ) Y. Furthermore F(0) = log(exp X ) = X , and 1 F(1) = F(0) + F (t)dt, 0 hence 1 log(exp X exp Y ) = X + Exp(ad X ) Exp(t ad Y ) Y dt. 4 (Campbell–Hausdorff formula) If √ 1 2), then 2 log(exp X exp Y ) = X + ∞ k=0 (−1)k k+1 E(k) X , Y < 1 2 log(2 − 1 (q1 + · · · + qk + 1) (ad X ) p1 (ad Y )q1 . . q1 ! . m! where, for k ≥ 1, E(k) = { p1 , q1 , .

The representation π1 of G on the quotient space V/W is called a quotient representation. The representation π is said to be irreducible if the only invariant subspaces are {0} and V. Two representations (π1 , V1 ) and (π2 , V2 ) are said to be equivalent if there exists an isomorphism A : V1 → V2 (A is an invertible linear map) such that Aπ1 (g) = π2 (g)A, for every g ∈ G. One says that A is an intertwinning operator or that A intertwins the representations π1 and π2 . A representation of a Lie algebra g on a vector space V is a linear map ρ : g → End(V) which is a Lie algebra morphism: ρ([X, Y ]) = [ρ(X ), ρ(Y )] = ρ(X )ρ(Y ) − ρ(Y )ρ(X ).

For X ∈ M(n, R) with X < R the functional calculus associates to the function f the matrix f (X ) = ∞ k=0 ak X k . 26 The exponential map The map f → f (X ) ∈ M(n, R) is an algebra morphism: it is linear and ( f 1 f 2 )(X ) = f 1 (X ) f 2 (X ). Let ∞ g(z) = bm z m m=1 be another power series, with g(0) = b0 = 0. The function f ◦ g is the sum of a power series in a neighbourhood of 0, ∞ f ◦ g(z) = cpz p. 2 If ∞ (∗) |bm | X m < R, m=1 then g(X ), f (g(X )) and ( f ◦ g)(X ) are well defined, and ( f ◦ g)(X ) = f g(X ) .

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