Unipotent group

A subgroup $ U $ of a linear algebraic group $ G $ consisting of unipotent elements (cf. Unipotent element). If $ G $ is identified with its image under an isomorphic imbedding in a group $ \mathop{\rm GL}\nolimits (V) $ of automorphisms of a suitable finite-dimensional vector space $ V $ , then a unipotent group is a subgroup contained in the set $$ \{ {g \in \mathop{\rm GL}\nolimits (V)} : {(1 - g) ^{n} = 0} \} , n = \mathop{\rm dim}\nolimits \ V, $$ of all unipotent automorphisms of $ V $ . Fixing a basis in $ V $ , one may identify $ \mathop{\rm GL}\nolimits (V) $ with the general linear group $ \mathop{\rm GL}\nolimits _{n} (K) $ , where $ K $ is an algebraically closed ground field; the linear group $ U $ is then also called a unipotent group. An example of a unipotent group is the group $ U _{n} (K) $ of all upper-triangular matrices in $ \mathop{\rm GL}\nolimits _{n} (K) $ with 1's on the main diagonal. If $ k $ is a subfield of $ K $ and $ U $ is a unipotent subgroup in $ \mathop{\rm GL}\nolimits _{n} (k) $ , then $ U $ is conjugate over $ k $ to some subgroup of $ U _{n} (k) $ . In particular, all elements of $ U $ have in $ V $ a common non-zero fixed vector, and $ U $ is a nilpotent group. This theorem shows that the unipotent algebraic groups are precisely the Zariski-closed subgroups of $ U _{n} (k) $ for varying $ n $ .

In any linear algebraic group $ H $ there is a unique connected normal unipotent subgroup $ R _{u} (H) $ ( the unipotent radical) with reductive quotient group $ H/R _{u} (H) $ ( cf. Reductive group). To some extent this reduces the study of the structure of arbitrary groups to a study of the structure of reductive and unipotent groups. In contrast to the reductive case, the classification of unipotent algebraic groups is at present (1992) unknown.

Every subgroup and quotient group of a unipotent algebraic group is again unipotent. If $ \mathop{\rm char}\nolimits \ K = 0 $ , then $ U $ is always connected; moreover, the exponential mapping $ \mathop{\rm exp}\nolimits : \ \mathfrak u \rightarrow U $ ( where $ \mathfrak u $ is the Lie algebra of $ U $ ) is an isomorphism of algebraic varieties; if $ \mathop{\rm char}\nolimits \ K = p > 0 $ , then there exist non-connected unipotent algebraic groups: e.g. the additive group $ \mathbf G _{a} $ of the ground field (which may be identified with $ U _{2} (K) $ ) is a $ p $ - group and so contains a finite unipotent group. In a connected unipotent group $ U $ there is a sequence of normal subgroups $ U = U _{1} \supset \dots \supset U _{s} = \{ e \} $ such that all quotients $ U _{i} /U _ {i + 1} $ are one-dimensional. Every connected one-dimensional unipotent algebraic group is isomorphic to $ \mathbf G _{a} $ . This reduces the study of connected unipotent algebraic groups to a description of iterated extensions of groups of type $ \mathbf G _{a} $ .

Much more is known about commutative unipotent algebraic groups (cf. [4]) than in the general case. If $ \mathop{\rm char}\nolimits \ K = 0 $ , then they are precisely the algebraic groups isomorphic to $ \mathbf G _{a} \times \dots \times \mathbf G _{a} $ ; here, the isomorphism $ \mathbf G _{a} \times \dots \times \mathbf G _{a} \rightarrow U $ is given by the exponential mapping. If $ \mathop{\rm char}\nolimits \ K = p > 0 $ , then the connected commutative unipotent algebraic groups $ U $ are precisely the connected commutative algebraic $ p $ - groups. Now $ U $ need not be isomorphic to $ \mathbf G _{a} \times \dots \times \mathbf G _{a} $ : for this it is necessary and sufficient that $ g ^{p} = e $ for all $ g \in U $ . In the general case $ U $ is isogenous (cf. Isogeny) to a product of certain special groups (so-called Witt groups, cf. [2]).

If $ H $ and $ U $ are connected unipotent algebraic groups and $ H \subset U $ , then the variety $ U/H $ is isomorphic to an affine space. Any orbit of a unipotent algebraic group of automorphisms of an affine algebraic variety $ X $ is closed in $ X $ [5].

References

[1]	A. Borel, "Linear algebraic groups" , Springer (1991) MR1102012 Zbl 0726.20030
[2]	J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) MR0103191
[3]	J.E. Humphreys, "Linear algebraic groups" , Springer (1981) MR0610979 MR0396773 Zbl 0471.20029
[4]	T. Kambayachi, M. Miyanishi, M. Takeuchi, "Unipotent algebraic groups" , Springer (1974) MR376696
[5]	R. Steinberg, "Conjugacy classes in algebraic groups" , Lect. notes in math. , 366 , Springer (1974) MR0352279 Zbl 0281.20037