Root system of a semi-simple Lie algebra
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|Group theory → Lie groups
In mathematics, there is a one-to-one correspondence between reduced crystallographic root systems and semi-simple Lie algebras. Here the construction of a root system of a semi-simple Lie algebra – and, conversely, the construction of a semi-simple Lie algebra from a reduced crystallographic root system – are shown.
Associated root system
Let g be a semi-simple complex Lie algebra. Let further h be a Cartan subalgebra of g. Then h acts on g via simultaneously diagonalizable linear maps in the adjoint representation. For λ in h* define the subspace gλ ⊂ g by
We call a non-zero λ in h* a root if the subspace gλ is nontrivial. In this case gλ is called the root space of λ. The definition of Cartan subalgebra guarantees that g0 = h. One can show that each non-trivial gλ (i.e. for λ≠0) is one-dimensional. Let R be the set of all roots. Since the elements of h are simultaneously diagonalizable, we have
The Cartan subalgebra h inherits an inner product from the Killing form on g. This induces an inner product on h*. One can show that with respect to this inner product R is a reduced crystallographic root lattice.
Associated semi-simple Lie algebra
with the Chevalley-Serre relations
[Here the coefficients denoted by should be replaced by the coefficients of the Cartan matrix.]
It turns out that the generated Lie algebra is semi-simple and has root system isomorphic to the given R.
Due to the isomorphism, classification of finite-dimensional representations of semi-simple Lie algebras is reduced to the somewhat easier task of classifying reduced crystallographic root systems.
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