Jordan–Chevalley decomposition

The Jordan - Chevalley decomposition (sometimes Dunford decomposition ) is a term from the mathematical subfield of Lie algebras. Was named the Jordan - Chevalley decomposition by Marie Ennemond Camille Jordan and Claude Chevalley. This decomposition is important for the study of Lie algebras and algebraic groups.

Under the ( additive ) Jordan - Chevalley decomposition of an endomorphism of a finite dimensional vector space over an algebraically closed field is defined as the sum which a semisimple (ie diagonalisierbarer ) and a nilpotent endomorphism are, the commute, that is.

Is general a semisimple Lie algebra ( with Lie bracket ) over an algebraically closed field of characteristic 0 and so is referred to as (additive abstract ) Jordan - Chevalley decomposition if the following holds: The endomorphism is semisimple, the endomorphism is nilpotent, and it is true. This mapping is defined as follows for each:

Which is an endomorphism of.

The Jordan - Chevalley decomposition exists in the cases mentioned above and is unique. In addition, both definitions agree in the case, provided with the Lie bracket match.

The multiplicative decomposition is an invertible operator dar. as a product of its commuting semisimple and unipotent shares