An Introduction to Affine Kac Moody Lie Algebras and Groups
The collection of indecomposable Kac-Moody algebras is divided into three mutually exclusive types: finite, affine, and indefinite. The finite type indecomposable Kac-Moody algebras are precisely the finite-dimensional simple Lie algebras. The aim of this chapter is to explicitly realize the Kac-Moody algebras of affine type (also called the affine Kac-Moody algebras) and the associated groups. Most of the important applications of Kac-Moody theory so far center around this type. Actually, we only consider the Kac-Moody algebras g and the associated groups G of “untwisted” affine type. The “twisted” ones are obtained from the untwisted g as the subalgebra consisting of the elements fixed under an automorphism of g of finite order. The twisted affine Kac-Moody groups are obtained similarly.
KeywordsAlgebraic Group Weyl Group Parabolic Subgroup Root Vector Schubert Variety
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