Affine Lie Algebra

These study will lead us to wonder the reason why we encounter objects, such as modular forms, affine Lie algebras, the conformal held theory etc. Usually we think these objects live with elliptic curves, not with surfaces. Formally, we have two possibilities one is that these objects are so universal (like Dynkin diagrams) that they appear everywhere. The other possiblity is that elliptic curves are hidden in the Hilbert schemes. We do not know which is correct at this moment, but we believe that the second one is correct. 

LB. Frenkel, Spinor representations of affine Lie algebras, Proc. Natl. Acad. Sci. USA 77 (1980), 6303-6306. 

Reyman, A. G., and Semenov-Tian-Shansky, M. A. Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. Invent. Math. 54 (1979), No. 

Hochschild, G. Introduction to Affine Algebraic Groups (San Francisco Holden-Day, 1971). Mainly algebraic matrix groups, with Hopf-algebraic treatment. The emphasis is on characteristic zero and relation with Lie algebras. 

Theorem. Let kbe a field of characteristic zero. Let G be a connected affine algebraic group schemeacting linearly on V. A subspace W of Vis stable under G iff it is stable under Lie(G). 

Let G be an affine algebraic group scheme. Show that always dim Lie(G) > dim G. [Pass to k and note Lie(Gred) S Lie(G). A ring-theoretic proof is also possible 1. 

Corollaire 4.9 Soit G un groups algebrique sur un corps algebriquement clos k. On suppose G lisse sur k, connexe affine, a centre reductif r duit au groupe unit4, et que 1 algebra de Lie de G est nilpotente. ... 

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