Jordan decomposition (of an endomorphism)

The Jordan decomposition of an endomorphism of a finite-dimensional vector space is the representation of as the sum of a semi-simple and a nilpotent endomorphism that commute with each other: . The endomorphisms and are said to be the semi-simple and the nilpotent component of the Jordan decomposition of . This decomposition is called the additive Jordan decomposition. (A semi-simple endomorphism is one having a basis of eigen vectors for some extension of the ground field, a nilpotent endomorphism is one some power of which is the zero endomorphism.) If in some basis of the space the matrix of is a Jordan matrix (i.e., a matrix in Jordan canonical form), and is an endomorphism such that the matrix of in the same basis has for and for all , then

is the Jordan decomposition of with and .

The Jordan decomposition exists and is unique for any endomorphism of a vector space over an algebraically closed field . Moreover, and for some polynomials and over (depending on ) with constant terms equal to zero. If is a -invariant subspace of , then is invariant under and , and

is the Jordan decomposition of (here means restriction to ). If is a subfield of and is rational over (with respect to some -structure on ), then and need not be rational over ; one may only assert that and are rational over , where is the characteristic exponent of (for , is , and for it is the set of all elements of that are purely inseparable over .

If is an automorphism of , then is also an automorphism of , and

where and is the identity automorphism of . The automorphism is unipotent, that is, all its eigen values are equal to one. Every representation of as a product of commuting semi-simple and unipotent automorphisms coincides with the representation already described. This representation is called the multiplicative Jordan decomposition of the automorphism , and and are called the semi-simple and unipotent components of . If is rational over , then and are rational over . If is a -invariant subspace of , then is invariant under and , and

is the multiplicative Jordan decomposition of .

The concept of a Jordan decomposition can be generalized to locally finite endomorphisms of an infinite-dimensional vector space , that is, endomorphisms such that is generated by finite-dimensional -invariant subspaces. For such , there is one and only one decomposition of as a sum (and in the case of an automorphism, one and only one decomposition of as a product ) of commuting locally finite semi-simple and nilpotent endomorphisms (semi-simple and unipotent automorphisms, respectively), that is, endomorphisms (automorphisms) such that every finite-dimensional -invariant subspace of is invariant under and ( and , respectively) and (, respectively) is the Jordan decomposition of .

This extension of the concept of a Jordan decomposition allows one to introduce the concept of a Jordan decomposition in algebraic groups and algebras. Let be an affine algebraic group over (cf. Affine group), let be its Lie algebra, let be the representation of in the group of automorphisms of the algebra of regular functions on defined by right translations, and let be its derivation. For arbitrary in and in , the endomorphisms and of the vector space are locally finite, so that one can speak of their Jordan decompositions:

and

One of the important results in the theory of algebraic groups is that the Jordan decomposition just indicated is realized by the use of elements of and , respectively. More exactly, there exist unique elements and such that

(1)

(2)

and

The decomposition (1) is called the Jordan decomposition in the algebraic group , and (2) the Jordan decomposition in the algebraic Lie algebra . If is defined over a subfield of and the element (, respectively) is rational over , then and ( and , respectively) are rational over . Moreover, if is realized as a closed subgroup of the general linear group of automorphisms of some finite-dimensional vector space (and thus is realized as a subalgebra of the Lie algebra of ), then the Jordan decomposition (1) of coincides with the multiplicative decomposition introduced above for , while the decomposition (2) for coincides with the additive Jordan decomposition for (considered as endomorphisms of ). If is a rational homomorphism of affine algebraic groups and is the corresponding homomorphism of their Lie algebras, then

for arbitrary , .

The concept of a Jordan decomposition in algebraic groups and algebraic Lie algebras allows one to introduce the definitions of a semi-simple and a unipotent (nilpotent, respectively) element in an arbitrary affine algebraic group (algebraic Lie algebra, respectively). An element is said to be semi-simple if , and unipotent if ; an element is said to be semi-simple if and nilpotent if . If is defined over , then

is a -closed subset of , and

is a -closed subset of . In general,

is not a closed set, but if is commutative, then and are closed subgroups and . The sets and in an arbitrary affine algebraic group are invariant under inner automorphisms, and the study of decompositions of these sets into classes of conjugate elements is a subject of special investigations [3].

References

[1]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2]	E.R. Kolchin, "Algebraic matrix groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations" Ann. of Math. , 49 (1948) pp. 1–42
[3]	A. Borel (ed.) R. Carter (ed.) C.W. Curtis (ed.) N. Iwahori (ed.) T.A. Springer (ed.) R. Steinberg (ed.) , Seminar on algebraic groups and related finite groups , Lect. notes in math. , 131 , Springer (1970) Zbl 0192.36201

V.L. Popov

References

[a1]	R. Steinberg, "Conjugacy classes in algebraic groups" , Lect. notes in math. , 366 , Springer (1974) MR0352279 Zbl 0281.20037