Lie Algebra Root

The roots of a semisimple Lie algebra g are the Lie algebra weights occurring in its adjoint representation. The set of roots form the root system, and are completely determined by g. It is possible to choose a set of Lie algebra positive roots, every root alpha is either positive or -alpha is positive. The Lie algebra simple roots are the positive roots which cannot be written as a sum of positive roots.

The simple roots can be considered as a linearly independent finite subset of Euclidean space, and they generate the root lattice. For example, in the special Lie algebra sl_2C of two by two matrices with zero matrix trace, has a basis given by the matrices

 H=[1  0; 0 -1],X=[0 1; 0 0],Y=[0 0; 1 0].

The adjoint representation is given by the brackets


so there are two roots of sl_2 given by alpha(H)=2 and -alpha(H)=-2. The Lie algebraic rank of sl_2C is one, and it has one positive root.

See also

Cartan Matrix, Lie Algebra, Lie Algebra Weight, Semisimple Lie Algebra, Weyl Group

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Lie Algebra Root." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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