Root system of a semi-simple Lie algebra
In mathematics, there is a one-to-one correspondence between reduced crystallographic root systems and semisimple Lie algebras. Here the construction of a root system of a semisimple Lie algebra – and, conversely, the construction of a semisimple Lie algebra from a reduced crystallographic root system – are shown.
|Group theory → Lie groups|
Associating a root system to a semisimple Lie algebra
Let be a complex semisimple Lie algebra. Let further be a Cartan subalgebra of . Then acts on via simultaneously diagonalizable linear maps in the adjoint representation. For λ in define the subspace by
We say that is a root if and the subspace is nonzero. In this case is called the root space of λ. For each root , the root space is one-dimensional. Meanwhile, the definition of Cartan subalgebra guarantees that .
Let R be the set of all roots. Since the elements of are simultaneously diagonalizable, we have
The Cartan subalgebra inherits a nondegenerate bilinear form from the Killing form on . This form induces a form on and the restriction of that form to the real span of the roots is an inner product. One can show that with respect to this inner product R is a reduced crystallographic root system.
Associating a semisimple Lie algebra to a root system
Here is the coefficient of the Cartan matrix, given by
Note that if and are in with , then , so that is a non-positive integer and is a positive integer.
Finding generators with these relations
Now, if we are given a semisimple Lie algebra with root system , it is not particularly difficult to find a set of generators for satisfying the above relations. For each simple root , we can find in the root space , in the root space and in the Cartan subalgebra satisfying the standard relations: , , and . These will be our generators.
This sort of reasoning verifies the first four relations above.
The last two relations hold because belongs to the root space , and more generally, belongs to . But, as we shall see momentarily, if and are simple roots, then is not a root if , so that must be zero. To see that is not a root, note that if and are distinct elements of , then cannot be a root, for this would violate one of the defining properties of a base—that the expansion of a root in terms of the base cannot have both positive and negative coefficients. But then if is the reflection associated to , we can easily calculate that
Then since is not a root, neither is .
Serre's theorem asserts much more than just the existence of such generators for a given semisimple Lie algebra . First, the claim is that the above relations completely determine ; that is, there are no other relations in besides ones that follow from these. Even more, however, Serre's theorem asserts that starting from an arbitrary root system—not assumed to come from a semisimple Lie algebra—we can use the above relations to define a Lie algebra, the Lie algebra is finite-dimensional and semisimple, and the root system of that Lie algebra is the root system we started from.
A consequence of Serre's theorem is this:
- Every (reduced, crystallographic) root system comes from a semisimple Lie algebra.
Application to the classification of semisimple Lie algebras
The preceding results help in the process of reducing the classification of semisimple Lie algebras to the task of classifying reduced crystallographic root systems, which is then done in terms of Dynkin diagrams. Although the above describes how to construct a root system from a Lie algebra and vice versa, there are still two things to prove before we can obtain a one-to-one correspondence.
- First, we need to know that each Lie algebra gives only one root system (up to isomorphism). This is shown by proving that the Cartan subalgebra of a semisimple Lie algebra is unique up to automorphism. It follows that all Cartan subalgebras of give isomorphic root systems.
- Second, we must prove that each root system gives only one Lie algebra (up to isomorphism). That is, we must show that if two Lie algebras have isomorphic root systems, the Lie algebras are also isomorphic.
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