# Cartan theorem

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Cartan's theorem on the highest weight vector. Let be a complex semi-simple Lie algebra, let , , be canonical generators of it, that is, linearly-independent generators for which the following relations hold:  where , are non-positive integers when , , implies , and let be the Cartan subalgebra of which is the linear span of . Also let be a linear representation of in a complex finite-dimensional space . Then there exists a non-zero vector for which where the are certain numbers. This theorem was established by E. Cartan . The vector is called the highest weight vector of the representation and the linear function on defined by the condition , , is called the highest weight of the representation corresponding to . The ordered set is called the set of numerical marks of the highest weight . Cartan's theorem gives a complete classification of irreducible finite-dimensional linear representations of a complex semi-simple finite-dimensional Lie algebra. It asserts that each finite-dimensional complex irreducible representation of has a unique highest weight vector (up to proportionality), and that the numerical marks of the corresponding highest weight are non-negative integers. Two finite-dimensional irreducible representations are equivalent if and only if the corresponding highest weights are the same. Any set of non-negative integers is the set of numerical marks of the highest weight of some finite-dimensional complex irreducible representation.

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How to Cite This Entry:
Cartan theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan_theorem&oldid=15979
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article