By Hall B.C.
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Extra info for An Elementary Introduction to Groups and Representations
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This is a contradiction, because D is the orthogonal complement of g. So for every neighborhood U of zero in g, exp (U ) contains a neighborhood of the identity in G. If we make U small enough, then the exponential will be one-toone on U . ) Let log denote the inverse map, defined on exp U . Since U is compact, and exp is one-to-one and continuous on U , log will be continuous. ) So take V to be a neighborhood of I contained in exp U , and let U = exp−1 (V ) ∩ U . Then U is open and the exponential takes U homeomorphically onto V .
Then m2 lim m→∞ I+ X + Cm m m = eX . Proof. The expression inside the brackets is clearly tending to I as m → ∞, and so is in the domain of the logarithm for all sufficiently large m. 7 satisfies Em ≤ c const. But then m2 I+ X + Cm = exp m X m + Cm 2 ≤ X + Cm + Em , m and so I+ X + Cm m m = exp (X + mCm + mEm ) . Since both Cm and Em are of order m12 , we obtain the desired result by letting m → ∞ and using the continuity of the exponential. 34 3. LIE ALGEBRAS AND THE EXPONENTIAL MAPPING 4. Further Properties of the Matrix Exponential In this section we give three additional results involving the exponential of a matrix, which will be important in our study of Lie algebras.
It is straightforward to verify that the above definition really makes VC into a complex vector space. We will regard V as a real subspace of VC in the obvious way. 35. Let g be a finite-dimensional real Lie algebra, and g C its complexification (as a real vector space). Then the bracket operation on g has a unique extension to gC which makes gC into a complex Lie algebra. The complex Lie algebra gC is called the complexification of the real Lie algebra g. Proof. 18) [X1 + iX2 , Y1 + iY2 ] = ([X1 , Y1] − [X2 , Y2]) + i ([X1 , Y2] + [X2 , Y1]) .