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The affine Lie algebra corresponding to a finite-dimensional semisimple Lie algebra is the direct sum of the affine Lie algebras corresponding to its simple summands. There is a distinguished derivation of the affine Lie algebra defined by (+) = ().
There are extensions of Dynkin diagrams, namely the affine Dynkin diagrams; these classify Cartan matrices of affine Lie algebras. These are classified in ( Kac 1994 , Chapter 4, pp. 47– ), specifically listed on ( Kac 1994 , pp. 53–55 ).
This article gives a table of some common Lie groups and their associated Lie algebras.. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple).
Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.)
Ado's theorem: Any finite-dimensional Lie algebra is isomorphic to a subalgebra of for some finite-dimensional vector space V. affine 1. An affine Lie algebra is a particular type of Kac–Moody algebra. 2. An affine Weyl group. analytic 1. An analytic subgroup automorphism 1.
The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple ...
Central extensions of a Lie algebra g by an abelian Lie algebra h can be obtained with the help of a so-called (nontrivial) 2-cocycle on g. Non-trivial 2-cocycles occur in the context of projective representations of Lie groups. This is alluded to further down. A Lie algebra extension
Finite type indecomposable matrices classify the finite dimensional simple Lie algebras (of types ,,,,,), while affine type indecomposable matrices classify the affine Lie algebras (say over some algebraically closed field of characteristic 0).