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# Lie Algebra Root

The roots of a semisimple Lie algebra are the Lie algebra weights occurring in its adjoint representation. The set of roots form the root system, and are completely determined by . It is possible to choose a set of Lie algebra positive roots, every root is either positive or 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 of two by two matrices with zero matrix trace, has a basis given by the matrices

 (1)

The adjoint representation is given by the brackets

 (2)
 (3)

so there are two roots of given by and . The Lie algebraic rank of is one, and it has one positive root.

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

This entry contributed by Todd Rowland

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Rowland, Todd. "Lie Algebra Root." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LieAlgebraRoot.html