Jordan–Chevalley decomposition

In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley (also known as Dunford decomposition, named after Nelson Dunford, as well as SN decomposition), expresses a linear operator as the sum of its commuting semisimple part and its nilpotent parts. The multiplicative decomposition expresses an invertible operator as the product of its commuting semisimple and unipotent parts. The decomposition is important in the study of algebraic groups. The decomposition is easy to describe when the Jordan normal form of the operator is given, but it exists under weaker hypotheses than the existence of a Jordan normal form.

Linear operators

Consider linear operators on a finite-dimensional vector space over a perfect field. An operator is semisimple if the roots (in an extension of the ground field) of its minimal polynomial are all distinct (if the underlying field is algebraically closed, this is the same as the requirement that the operator be diagonalizable). An operator x is nilpotent if some power x^m of it is the zero operator. An operator x is unipotent if x − 1 is nilpotent.

Now, let x be any operator. A Jordan–Chevalley decomposition of x is an expression of it as a sum:

x = x_ss + x_n,

where x_ss is semisimple, x_n is nilpotent, and x_ss and x_n commute. If such a decomposition exists it is unique, and x_ss and x_n are in fact expressible as polynomials in x, (Humphreys 1972, Prop. 4.2, p. 17).

If x is an invertible operator, then a multiplicative Jordan–Chevalley decomposition expresses x as a product:

x = x_ss · x_u,

where x_ss is semisimple, x_u is unipotent, and x_ss and x_u commute. Again, if such a decomposition exists it is unique, and x_ss and x_u are expressible as polynomials in x.

For endomorphisms of a finite dimensional vector space whose characteristic polynomial splits into linear factors over the ground field (which always happens if that is an algebraically closed field), the Jordan–Chevalley decomposition exists and has a simple description in terms of the Jordan normal form. If x is in the Jordan normal form, then x_ss is the endomorphism whose matrix on the same basis contains just the diagonal terms of x, and x_n is the endomorphism whose matrix on that basis contains just the off-diagonal terms; x_u is the endomorphism whose matrix is obtained from the Jordan normal form by dividing all entries of each Jordan block by its diagonal element.

If the ground field is not perfect, then a Jordan–Chevalley decomposition may not exist. Example: Let p be a prime number, let F = F_p (X^p), let V = F_p (X). This is an extension field of vector space dimension p. Let x be the multiplication by the indeterminate X; it is an endomorphism of V. It is easy to see that the minimal polynomial of x is (t^p − X^p), which has X as the only root in F_p (X) and so x is not semisimple. Therefore, if a Jordan–Chevalley decomposition would exist, the nilpotent part would have to be nonzero. But since the nilpotent part is also a polynomial in x, it belongs to the field F_p (X) and therefore must be zero, which gives a contradiction. Hence no Jordan–Chevalley decomposition exists in this example.

Banach spaces

For operators on Banach spaces, there is a decomposition known as the Dunford decomposition which generalizes the Jordan–Chevalley decomposition, (Dunford 1946).

References

• Dunford, Nelson (1946), "Direct decompositions of Banach spaces", Boletí n de la Sociedad Matemática Mexicana 3: 1–12, MR0021240 
• Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Springer, ISBN 978-0-387-90053-7 , p 559.
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR1878556