Jordan-Chevalley decomposition

The Jordan – Chevalley decomposition (sometimes also the Dunford decomposition) is important for studying Lie algebras and algebraic groups . It is named after Marie Ennemond Camille Jordan and Claude Chevalley .

The (additive) Jordan-Chevalley decomposition of an endomorphism of a finite-dimensional vector space over an algebraically closed field is understood to be the sum in which a semi-simple (i.e. diagonalizable ) and a nilpotent endomorphism are, which commute with each other, that is . ${\displaystyle x\colon V\rightarrow V}$ ${\displaystyle V}$ ${\displaystyle x=x_{s}+x_{n}}$ ${\displaystyle x_{s}}$ ${\displaystyle x_{n}}$ ${\displaystyle x_{s}x_{n}=x_{n}x_{s}}$

If, more generally, a semi-simple Lie algebra (with Lie brackets ) over an algebraically closed field of the characteristic 0 and , then one calls an (additive abstract) Jordan-Chevalley decomposition, if the following applies: The endomorphism is semi-simple, the endomorphism is nilpotent, and it applies . It defines the mapping for each as follows: ${\displaystyle L}$ ${\displaystyle [\cdot ,\cdot ]}$ ${\displaystyle x\in L}$ ${\displaystyle x=x_{s}+x_{n}}$ ${\displaystyle {\rm {ad}}(x_{s})}$ ${\displaystyle {\rm {ad}}(x_{n})}$ ${\displaystyle [x_{s},x_{n}]=0}$ ${\displaystyle y\in L}$ ${\displaystyle {\rm {ad}}(y)}$

${\displaystyle {\rm {ad}}(y):L\rightarrow L\ ,\ z\mapsto [y,z]}$,

which is an endomorphism of . ${\displaystyle L}$

The Jordan-Chevalley decomposition exists in the cases given above and is unique. In addition, both definitions match in the case provided with the Lie bracket . ${\displaystyle L={\rm {End}}(V)}$ ${\displaystyle [f,g]:=fg-gf}$

The multiplicative decomposition represents an invertible operator as the product of its commuting semi-simple and unipotent components. This can easily be obtained from the additive decomposition given above:

${\displaystyle x=x_{s}+x_{n}=x_{s}\cdot (1+x_{s}^{-1}x_{n})}$.

Note that it is invertible because , as an invertible endomorphism, it cannot have the eigenvalue 0, and that because of the interchangeability of the factors it is also nilpotent and thus unipotent. ${\displaystyle x_{s}}$ ${\displaystyle x}$ ${\displaystyle x_{s}^{-1}x_{n}}$ ${\displaystyle 1+x_{s}^{-1}x_{n}}$

See also

• Jordan normal form

literature

• Serge Lang , Algebra (3 ed) , Addison-Wesley , 1993, ISBN 0-201-55540-9 . Chap.XIV.2, p.559.

Web links

• Jordan-Chevalley decomposition and Cartan criterion (PDF file; 178 kB)