Lie Algebra -- from Wolfram MathWorld

A nonassociative algebra obeyed by objects such as the Lie bracket and Poisson bracket. Elements , , and of a Lie algebra satisfy

and

(the Jacobi identity). The relation implies

For characteristic not equal to two, these two relations are equivalent.

The binary operation of a Lie algebra is the bracket

An associative algebra with associative product can be made into a Lie algebra by the Lie product

Every Lie algebra is isomorphic to a subalgebra of some where the associative algebra may be taken to be the linear operators over a vector space (the Poincaré-Birkhoff-Witt theorem; Jacobson 1979, pp. 159-160). If is finite dimensional, then can be taken to be finite dimensional (Ado's theorem for characteristic ; Iwasawa's theorem for characteristic ).

The classification of finite dimensional simple Lie algebras over an algebraically closed field of characteristic 0 can be accomplished by (1) determining matrices called Cartan matrices corresponding to indecomposable simple systems of roots and (2) determining the simple algebras associated with these matrices (Jacobson 1979, p. 128). This is one of the major results in Lie algebra theory, and is frequently accomplished with the aid of diagrams called Dynkin diagrams.

See also

Ado's Theorem, Derivation Algebra, Dynkin Diagram, Iwasawa's Theorem, Jacobi Identities, Lie Algebroid, Lie Bracket, Lie Group, Poincaré-Birkhoff-Witt Theorem, Poisson Bracket, Reduced Root System, Root System, Weyl Group Explore this topic in the MathWorld classroom

Explore with Wolfram|Alpha

References

Humphreys, J. E. Introduction to Lie Algebras and Representation Theory, 3rd ed. New York: Springer-Verlag, 1977.Jacobson, N. Lie Algebras. New York: Dover, 1979.Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 3, 1996.Weisstein, E. W. "Books about Lie Algebra." http://www.ericweisstein.com/encyclopedias/books/LieAlgebra.html.

Referenced on Wolfram|Alpha

Lie Algebra

Cite this as:

Weisstein, Eric W. "Lie Algebra." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LieAlgebra.html

Subject classifications