By Werner Hildbert Greub, Stephen Halperin, James Van Stone

ISBN-10: 0123027039

ISBN-13: 9780123027030

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Then a ⊂ p, and [a, p] ⊂ a ∩ [p, p] ⊂ a ∩ k = 0, and hence a ⊂ z(p). This implies that a = 0. Therefore, g is semisimple. 27. A reduced orthogonal Lie algebra (g, σ ) is maximal if for any reduced orthogonal involutive Lie algebra (g , σ ) such that g ⊃ g, σ restricts to σ on g and p = p, the equality g = g holds. 28. Let (g, σ ) be a reduced semisimple orthogonal involutive semisimple Lie algebra. Then it is maximal and k = [p, p]. Proof. Let (g , σ ) ⊃ (g, σ ) be a reduced orthogonal involutive Lie algebra such that σ restricts to σ and p = p.

Then they also form a basis of gC over C. For any X, Y ∈ g, ad X ad Y maps g to g and ig to ig. This implies immediately that the Killing form B(X, Y ) = T r (ad X ad Y ) restricts to the Killing form of g. Similarly, since g is a real form of gC , B|g is also the Killing form of g . Now for X ∈ p, i X ∈ p , and B(i X, i X ) = −B(X, X ), and hence if B = cQ on p, then B = −cQ on p . This implies that (g, σ ) is of compact type if and only if (g , σ ) is of noncompact type. The above lemma shows that to study irreducible reduced orthogonal involutive Lie algebras (g, σ ), it suffices to study those of noncompact type.

The existence follows from Zorn’s lemma. In fact, the set of compactifications 1 2 of X which are common quotients of X and X is partially ordered. It can be shown that every chain has a maximal element, and Zorn’s lemma implies the existence of a maximal common quotient. The existence of LCR can be proved similarly. 2] for more details. 20. For X = D × D, the GC Q of the Satake compactification X τ S and the conic compactification, X ∧ (X ∪ X (∞)), is the one point compactification X ∪ {∞}. Proof.

### Connections, curvature, and cohomology by Werner Hildbert Greub, Stephen Halperin, James Van Stone

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