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Universal central extensions Central trick
The action Ad Structure of g_ 2 2
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1-dimensional G-submodule 2-cocycle 2-connected acts trivially Ad(x admissible lattice algebra g algebraic group assume Aut(G automorphism central trick Chevalley group choose commutative component composition factors composition series contains Corollary corresponding defining relations degenerate sums denote direct sum Dynkin diagram elements exact sequence follows functor fundamental weight G-module group G Hence highest weight homomorphism indecomposable infinitesimally central extension Int w*(t invariant inverse image irreducible isomorphism k-extension k-homomorphism k-split ker tt kernel Lemma Let G Lie algebra linear long root maximal torus module morphism multiplicity non-zero norming constants notations orbits of degenerate p]-structure Proposition 11.18 Proposition 5.2 prove REMARK representation root system satisfies short root simple roots simply connected spanned Steinberg subgroup subspace suppose surjective T-equivariant type B3 type F unipotent radical universal central extension vector space weight space Weyl group zero