Coroot

From Wikipedia, the free encyclopedia

The introduction to this article provides insufficient context for those unfamiliar with the subject. Please help improve the article with a good introductory style.

In mathematics, in the field of representation theory of Lie algebras, a coroot is a certain kind of element of a Cartan subalgebra of a complex semisimple Lie algebra g.

The structure and representation theory of g is characterised by its root system. Given a root, α of a g, there are associated to it two operators;

X_α

and

Y_α,

known as the raising and lowering operators respectively.

Their Lie bracket,

H_α = [X_α,Y_α]

is an element of the Cartan subalgebra.

These raising and lowering operators are determined only up to scalar multipliers. It is often useful to set their lengths so as to form a subalgebra isomorphic to

sl(2, C),

the Lie algebra of the special linear group, of dimension 3.

Once this has been done

H_α

is the coroot associated to α (French copoid).