Coroot

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;

:$X\_\{alpha\}$

and

:$Y\_\{alpha\},$

known as the raising and lowering operators respectively.

Their Lie bracket,

:$H\_\{alpha\}\; =\; [X\_\{alpha\},\; Y\_\{alpha\}]$ 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

:$H\_\{alpha\}$

is the "coroot" associated to α (French "copoid").