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The Classical Simple Groups
Simple Lie Algebras
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abstract Coxeter complex AT-character Cartan decomposition Cartan subalgebra chamber complex Chevalley basis Chevalley group classical groups co-roots coefficients consider contains corresponding Coxeter element Coxeter group decomposition defined denote double coset Dynkin diagram eigenvalues element of G equivalence classes field finite follows fundamental roots fundamental system gallery given group G Hence hyperplane identity induction integer isometry isomorphic leaves invariant Lemma linear combination linearly independent matrices morphism multiplication negative roots nilpotent non-zero normal subgroup obtain operates orthogonal groups p-orbits pair parabolic subgroups permutation positive roots Proof Proposition prove r-chain reII relations root system root vectors satisfying scalar product self-conjugate simple groups simple Lie algebra structure constants subgroup of G subset subspace Suppose symmetric symplectic symplectic group system of type Theorem transforms twisted groups type G2 type-preserving automorphisms unique Weyl group Xr(t