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Symmetry And Group

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

By Hall B.C.

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1), e φ eX eY = eφ(log(e X Y e )) . 3) tells us that φ eX eY = e e e log eφ(X) eφ(Y ) e e = eφ(X) eφ(Y ) = φ(eX )φ(eY ). Thus, the Baker-Campbell-Hausdorff formula shows that on elements of the form eX , with X small, φ is a group homomorphism. ) The Baker-Campbell-Hausdorff formula shows that all the information about the group product, at least near the identity, is “encoded” in the Lie algebra. , to be a group homomorphism. In this section we will look at how all of this works out in the very special case of the Heisenberg group.

X) = dt φ(etX ), for all X ∈ g. t=0 If G, H, and K are matrix Lie groups and φ : H → K and ψ : G → H are Lie group homomorphisms, then φ ◦ ψ = φ ◦ ψ. In practice, given a Lie group homomorphism φ, the way one goes about computing φ is by using Property 3. Of course, since φ is (real) linear, it suffices to compute φ on a basis for g. In the language of differentiable manifolds, Property 3 says that φ is the derivative (or differential) of φ at the identity, which is the standard definition of φ.

Well, A can be written as eX for a unique X ∈ h and B can be written as eY for a unique Y ∈ h. 1 1 φ (AB) = φ eX eY = φ eX+Y + 2 [X,Y ] . Using the definition of φ and the fact that φ is a Lie algebra homomorphism: φ (AB) = exp φ (X) + φ (Y ) + 1 φ (X) , φ (Y ) 2 . 1 again we have e e φ (AB) = eφ(X) eφ(Y ) = φ (A) φ (B) . Thus φ is a group homomorphism. It is easy to check that φ is continuous (by checking that log, exp, and φ are all continuous), and so φ is a Lie group homomorphism. Moreover, φ by definition has the right relationship to φ.

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