Symmetry And Group

Analysis on Lie Groups - Jacques Faraut by Jacques Faraut.

By Jacques Faraut.

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Extra resources for Analysis on Lie Groups - Jacques Faraut

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This fact is at the origin of the notion of a Lie algebra. 1 Exponential of a matrix The exponential of a matrix X ∈ M(n, K) (K = R or C) is defined as the sum of the series exp(X ) = ∞ k=0 Xk . k! Since X k ≤ X k , the series converges normally for every matrix X , and uniformly on any compact set in M(n, K). If X and Y commute, X Y = Y X , then exp(X + Y ) = exp X exp Y . It follows that exp(X ) is invertible, and (exp X )−1 = exp(−X ). For g ∈ G L(n, K), g exp Xg −1 = exp(g Xg −1 ). If X is diagonalisable: λ  1 X = g ..

If π1 and π2 are two equivalent representations, then the derived representations dπ1 and dπ2 are also equivalent. The converse holds if G is connected.

We can write ∞ g(z)k = bm,k z m , m=k where bm,k = bm 1 bm 2 . . bm k , m 1 +···+m k =m and ∞ |bm,k |r m ≤ m=k k ∞ |bm |r m m=1 Then f ◦ g(z) = ∞ m=0 cm z m , . 2 Logarithm of a matrix 27 with m cm = ak bm,k , k=0 and ∞ |cm |r m ≤ m=0 ≤ ∞ m m=0 k=0 ∞ |ak ||bm,k | r m = ∞ |ak | |bm,k |r m m=k k=0 ∞ |ak | k=0 ∞ k |bm |r m . m=1 Assume that ∞ |bm | X m < R, m=1 then the series ∞ |cm | X m m=0 converges and ( f ◦ g)(X ) = ∞ cm X m = m=0 ∞ m ak bm,k m=0 k=0 Since ∞ m |ak ||bm,k | X m < ∞, m=0 k=0 one can invert the summations ( f ◦ g)(X ) = = ∞ ∞ bm,k X m ak k=0 m=k ∞ ∞ ak k=0 k bm X m .

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