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

By Hall B.C.

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Let G be a matrix Lie group, with Lie algebra g. Then for each A ∈ G, AdA is an invertible linear transformation of g with inverse AdA −1, and Ad : G → GL(g) is a group homomorphism. Proof. Easy. 15 guarantees that AdA(X) is actually in g for all X ∈ g. Since g is a real vector space with some dimension k, GL(g) is essentially the same as GL(k; R). Thus we will regard GL(g) as a matrix Lie group. It is easy to show that Ad : G → GL(g) is continuous, and so is a Lie group homomorphism. , from g to gl(g), with the property that f eAdX = Ad eX .

A = ± 1 + c2 ) is a hyperbola. 8. The group SU(2). 9) A= α −β β α is in SU(2). 9) 2 2 for a unique pair (α, β) satisfying |α| + |β| = 1. (Thus SU(2) can be thought of as the three-dimensional sphere S 3 sitting inside C2 = R4. ) 9. The groups Sp (1; R), Sp (1; C), and Sp (1). Show that Sp (1; R) = SL (2; R), Sp (1; C) = SL (2; C), and Sp(1) = SU(2). 10. The Heisenberg group. Determine the center Z(H) of the Heisenberg group H. Show that the quotient group H/Z(H) is abelian. 11. Connectedness of SO(n).

Let G be a matrix Lie group, and g its Lie algebra. Let A(t) be a smooth d A(t). Show that X ∈ g. curve lying in G, with A(0) = I. 8. Note: This shows that the Lie algebra g coincides with what would be called the tangent space at the identity in the language of differentiable manifolds. Consider the space gl(n; C) of all n × n complex matrices. As usual, for X ∈ gl(n; C), define adX : gl(n; C) → gl(n; C) by adX(Y ) = [X, Y ]. Suppose that X is a diagonalizable matrix. Show, then, that adX is diagonalizable as an operator on gl(n; C).