By Karl-Heinz Fieseler

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Extra resources for Lie Groups [Lecture notes]

Sample text

10. Let g be a Lie algebra. 62 1. A (finite dimensional) g-module V is a (finite dimensional) K-vector space V together with a bilinear map g × V −→ V, (X, v) → Xv, such that [X, Y ]v = X(Y v) − Y (Xv). 2. A g-module is called (a) irreducible or simple if V and {0} are the only g-submodules. e. such that V = U ⊕ W . 3. A Lie algebra g is called semisimple, if all its finite dimensional gmodules are semisimple. So g-modules are in one-to-one correspondence with K-vector spaces V together with a Lie algebra homomorphism g −→ gl(V ).

V if γ( nk ) ∈ U resp. γ( nk ) ∈ V . That is possible, since U ∩ V is connected. βn ) with β1 := γ1 α1−1 , βk := αk−1 γk αk−1 , 2 ≤ k < n and βn := αn−1 γn . Since both U and V are simply connected and βk ([0, 1]) ⊂ U or βk ([0, 1]) ⊂ V , we get βk ∼ x0 , 1 ≤ k ≤ n, and thus γ ∼ x0 . 10. Let π : X −→ Y be a covering and ϕ : Z −→ Y be a continuous map from the simply connected topological space Z to Y . Then given points z0 ∈ Z, x0 ∈ X with ϕ(z0 ) = π(x0 ), there is a unique lifting ϕˆ : Z −→ X of ϕ with ϕ(z ˆ 0 ) = x0 .

Back to Lie algebras! Note first that, given an ideal a ⊂ g of a Lie algebra g we can endow the factor vector space with a natural Lie bracket [X + a, Y + a] := [X, Y ] + a, the resulting Lie algebra being called the factor algebra g/a of g mod(ulo) the ideal a. Furthermore that g = aD if dim g = dim a + 1 and D = ad(X) for some X ∈ g \ a. 3. The Lie algebra g is called solvable if there is a finite sequence of subalgebras g0 = 0 ⊂ g1 ⊂ ... ⊂ gr = g, such that gi ⊂ gi+1 is an ideal of gi+1 for i < r with abelian factor algebra gi+1 /gi (or equivalently [gi+1 , gi+1 ] ⊂ gi ).

### Lie Groups [Lecture notes] by Karl-Heinz Fieseler

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