By Hall B.C.

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52 3. LIE ALGEBRAS AND THE EXPONENTIAL MAPPING 14. 15. 16. 17. 18. 19. 20. Hint: Recall that Pascal’s Triangle gives a relationship between things of the form n+1 and things of the form nk . k The complexification of a real Lie algebra. Let g be a real Lie algebra, g C its complexification, and h an arbitrary complex Lie algebra. Show that every real Lie algebra homomorphism of g into h extends uniquely to a complex Lie algebra homomorphism of gC into h. (This is the universal property of the complexification of a real Lie algebra.

2.

Recall that the norm of a vector x in Cn is defined to be x = 2 |xi | . x, x = This norm satisfies the triangle inequality x+y ≤ x + y . The norm of a matrix A is defined to be A = sup x=0 Ax . x Equivalently, A is the smallest number λ such that Ax ≤ λ x for all x ∈ Cn . It is not hard to see that for any n × n matrix A, A is finite. 3) AB ≤ A B A+B ≤ A + B . It is also easy to see that a sequence of matrices Am converges to a matrix A if and only if Am − A → 0. ) A sequence of matrices Am is said to be a Cauchy sequence if Am − Al → 0 2 2 as m, l → ∞.

### An Elementary Introduction to Groups and Representations by Hall B.C.

by Charles

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