Jordan decomposition (of an endomorphism)

The Jordan decomposition of an endomorphism $ g $ of a finite-dimensional vector space is the representation of $ g $ as the sum of a semi-simple and a nilpotent endomorphism that commute with each other: $ g = g _{s} + g _{n} $. The endomorphisms $ g _{s} $ and $ g _{n} $ are said to be the semi-simple and the nilpotent component of the Jordan decomposition of $ g $. 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 $ \| a _{ij} \| $ of $ g $ is a Jordan matrix (i.e., a matrix in Jordan canonical form), and $ t $ is an endomorphism such that the matrix $ \| b _{ij} \| $ of $ t $ in the same basis has $ b _{ij} = 0 $ for $ i \neq j $ and $ b _{ii} = a _{ii} $ for all $ i $, then

$$ g \ = \ t + ( g - t ) $$

is the Jordan decomposition of $ g $ with $ g _{s} = t $ and $ g _{n} = g -t $.

The Jordan decomposition exists and is unique for any endomorphism $ g $ of a vector space $ V $ over an algebraically closed field $ K $. Moreover, $ g _{s} = P (g) $ and $ g _{n} = Q (g) $ for some polynomials $ P $ and $ Q $ over $ K $ (depending on $ g $) with constant terms equal to zero. If $ W $ is a $ g $-invariant subspace of $ V $, then $ W $ is invariant under $ g _{s} $ and $ g _{n} $, and

$$ g \mid _{W} \ = \ \left . g _{s} \right | _{W} + \left . g _{n} \right | _{W} $$

is the Jordan decomposition of $ g \mid _{W} $ (here $ \mid _{W} $ means restriction to $ W $). If $ k $ is a subfield of $ K $ and $ g $ is rational over $ k $ (with respect to some $ k $-structure on $ V $), then $ g _{s} $ and $ g _{n} $ need not be rational over $ k $; one may only assert that $ g _{s} $ and $ g _{n} $ are rational over $ k ^ {p ^ {- \infty}} $, where $ p $ is the characteristic exponent of $ k $ (for $ p = 1 $, $ k ^ {p ^ {- \infty}} $ is $ k $, and for $ p > 1 $ it is the set of all elements of $ K $ that are purely inseparable over $ k $.

If $ g $ is an automorphism of $ V $, then $ g _{s} $ is also an automorphism of $ V $, and

$$ g = g _{s} g _{u} \ = \ g _{u} g _{s} , $$

where $ g _{u} = 1 _{V} + g _ s^{-1} g _{n} $ and $ 1 _{V} $ is the identity automorphism of $ V $. The automorphism $ g _{u} $ is unipotent, that is, all its eigen values are equal to one. Every representation of $ g $ as a product of commuting semi-simple and unipotent automorphisms coincides with the representation $ g = g _{s} g _{u} = g _{u} g _{s} $ already described. This representation is called the multiplicative Jordan decomposition of the automorphism $ g $, and $ g _{s} $ and $ g _{u} $ are called the semi-simple and unipotent components of $ g $. If $ g $ is rational over $ k $, then $ g _{s} $ and $ g _{u} $ are rational over $ k ^ {p ^ {- \infty}} $. If $ W $ is a $ g $-invariant subspace of $ V $, then $ W $ is invariant under $ g _{s} $ and $ g _{u} $, and

$$ g \mid _{W} \ = \ \left . g _{s} \right | _{W} \left . g _{u} \right | _{W} $$

is the multiplicative Jordan decomposition of $ g \mid _{W} $.

The concept of a Jordan decomposition can be generalized to locally finite endomorphisms of an infinite-dimensional vector space $ V $, that is, endomorphisms $ g $ such that $ V $ is generated by finite-dimensional $ g $-invariant subspaces. For such $ g $, there is one and only one decomposition of $ g $ as a sum $ g = g _{s} + g _{n} $ (and in the case of an automorphism, one and only one decomposition of $ g $ as a product $ g _{s} g _{u} $) of commuting locally finite semi-simple and nilpotent endomorphisms (semi-simple and unipotent automorphisms, respectively), that is, endomorphisms (automorphisms) such that every finite-dimensional $ g $-invariant subspace $ W $ of $ V $ is invariant under $ g _{s} $ and $ g _{n} $ ($ g _{s} $ and $ g _{u} $, respectively) and $ g | _{W} = g _{s} \mid _{W} + g _{n} | _{W} $ ($ g \mid _{W} = g _{s} | _{W} g _{u} | _{W} $, respectively) is the Jordan decomposition of $ g \mid _{W} $.

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 $ G $ be an affine algebraic group over $ K $ (cf. Affine group), let $ {\mathcal G} $ be its Lie algebra, let $ \rho $ be the representation of $ G $ in the group of automorphisms of the algebra $ K [ G ] $ of regular functions on $ G $ defined by right translations, and let $ d \rho $ be its derivation. For arbitrary $ g $ in $ G $ and $ X $ in $ {\mathcal G} $, the endomorphisms $ \rho (g) $ and $ d \rho (X) $ of the vector space $ K [ G ] $ are locally finite, so that one can speak of their Jordan decompositions:

$$ \rho (g) \ = \ \rho (g) _{s} \rho (g) _{u} $$

and

$$ d \rho (X) \ = \ d \rho (X) _{s} + d \rho (X) _{n} . $$

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 $ G $ and $ {\mathcal G} $, respectively. More exactly, there exist unique elements $ g _{s} ,\ g _{u} \in G $ and $ X _{s} ,\ X _{n} \in {\mathcal G} $ such that

$$ \tag{1} g \ = \ g _{s} g _{u} \ = \ g _{u} g _{s} , $$

$$ \tag{2} X \ = \ X _{s} + X _{n} ,\ \ [ X _{s} ,\ X _{n} ] \ = \ 0 , $$

and

$$ \rho ( g _{s} ) \ = \ \rho (g) _{s} ,\ \ \rho ( g _{u} ) \ = \ \rho (g) _{u} , $$

$$ d \rho ( X _{s} ) \ = \ ( d \rho (X) ) _{s} ,\ \ d \rho ( X _{n} ) \ = \ ( d \rho (X) ) _{n} . $$

The decomposition (1) is called the Jordan decomposition in the algebraic group $ G $, and (2) the Jordan decomposition in the algebraic Lie algebra $ {\mathcal G} $. If $ G $ is defined over a subfield $ k $ of $ K $ and the element $ g \in G $ ($ X \in {\mathcal G} $, respectively) is rational over $ k $, then $ g _{s} $ and $ g _{u} $ ($ X _{s} $ and $ X _{n} $, respectively) are rational over $ k ^ {p ^ {- \infty}} $. Moreover, if $ G $ is realized as a closed subgroup of the general linear group $ \mathop{\rm GL}\nolimits (V) $ of automorphisms of some finite-dimensional vector space $ V $ (and thus $ {\mathcal G} $ is realized as a subalgebra of the Lie algebra of $ \mathop{\rm GL}\nolimits (V) $), then the Jordan decomposition (1) of $ g \in G $ coincides with the multiplicative decomposition introduced above for $ g $, while the decomposition (2) for $ X \in {\mathcal G} $ coincides with the additive Jordan decomposition for $ X $ (considered as endomorphisms of $ V $). If $ \phi : \ G _{1} \rightarrow G _{2} $ is a rational homomorphism of affine algebraic groups and $ d \phi : \ {\mathcal G} _{1} \rightarrow {\mathcal G} _{2} $ is the corresponding homomorphism of their Lie algebras, then

$$ \phi ( g _{s} ) \ = \ \phi (g) _{s} ,\ \ \phi ( g _{u} ) \ = \ \phi (g) _{u} , $$

$$ d \phi ( X _{s} ) \ = \ ( d \phi (X) ) _{s} ,\ \ d \phi ( X _{n} ) \ = \ ( d \phi (X) ) _{n} $$

for arbitrary $ g \in G _{1} $, $ X \in {\mathcal G} _{1} $.

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 $ g \in G $ is said to be semi-simple if $ g = g _{s} $, and unipotent if $ g = g _{u} $; an element $ X \in {\mathcal G} $ is said to be semi-simple if $ X = X _{s} $ and nilpotent if $ X = X _{n} $. If $ G $ is defined over $ k $, then

$$ G _{u} \ = \ \{ {g \in G} : {g = g _ u} \} $$

is a $ k $-closed subset of $ G $, and

$$ {\mathcal G} _{n} \ = \ \{ {X \in {\mathcal G}} : {X = X _ n} \} $$

is a $ k $-closed subset of $ {\mathcal G} $. In general,

$$ G _{s} \ = \ \{ {g \in G} : {g = g _ s} \} $$

is not a closed set, but if $ G $ is commutative, then $ G _{s} $ and $ G _{u} $ are closed subgroups and $ G = G _{s} \times G _{u} $. The sets $ G _{s} $ and $ G _{u} $ 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