Unipotent Groups
A unipotent group is a connected affine algebraic group whose elements act unipotently in every (equivalently, some) faithful linear representation. Unipotent groups are the "complementary" half of the structure theory: every connected linear algebraic group is an extension of a reductive group by its unipotent radical.
Definition
Let \(U\) be a connected affine algebraic group over a field \(k\). The following are equivalent:
- For every faithful representation \(U \hookrightarrow \mathrm{GL}_n\), every element of \(U(\bar k)\) is unipotent (\(I - g\) is nilpotent).
- There exists a closed embedding \(U \hookrightarrow \mathbb{U}_n\) into the strict upper-triangular subgroup of \(\mathrm{GL}_n\) for some \(n\).
- The Lie algebra \(\operatorname{Lie}(U)\) consists of nilpotent endomorphisms (in any faithful representation).
In characteristic \(0\), the exponential gives an iso of varieties \(\exp : \operatorname{Lie}(U) \xrightarrow{\sim} U\); in characteristic \(p\) this fails (Witt vectors enter).
Examples
| Group | Description |
|---|---|
| \(\mathbb{G}_a\) | Additive group, smallest unipotent. |
| \(\mathbb{U}_n\) | Strict upper-triangular \(n \times n\) matrices. |
| Heisenberg \(H_3\) | \(3 \times 3\) upper-unitriangular; non-commutative. |
| Witt vectors \(W_n\) | Char \(p\) phenomenon; not isomorphic to \(\mathbb{G}_a^n\). |
In char \(0\), every connected unipotent group is geometrically isomorphic to an affine space \(\mathbb{A}^n\) as a variety, but with a non-trivial group structure when non-commutative.
Kolchin's theorem
Theorem (Kolchin). Let \(V\) be a finite-dimensional vector space over an algebraically closed field \(k\), and let \(G \subseteq \mathrm{GL}(V)\) be a connected algebraic subgroup consisting entirely of unipotent elements. Then \(G\) stabilises a complete flag in \(V\) — i.e.\ \(G \subseteq \mathbb{U}_n\) for some basis.
In particular: every connected unipotent group is a closed subgroup of \(\mathbb{U}_n\) for some \(n\), and hence nilpotent as an abstract group.
Structure of commutative unipotent groups
For a commutative unipotent group \(U\) over a perfect field \(k\):
- Char \(0\): \(U \cong \mathbb{G}_a^n\) (as algebraic groups).
- Char \(p\): \(U\) admits a finite filtration whose successive quotients are subgroups of \(\mathbb{G}_a\) or of the Witt vector groups \(W_n\). The classification involves the Dieudonné module of \(U\).
A commutative unipotent group is \(p\)-torsion in characteristic \(p\); in characteristic \(0\) it is torsion-free and divisible.
Non-commutative unipotent groups
Examples: the Heisenberg group, the upper-triangular unitary group \(\mathbb{U}_n\), and the unipotent radicals of all proper parabolic subgroups of reductive groups. The composition factors of any unipotent group are isomorphic to \(\mathbb{G}_a\) (char \(0\)) or subgroups of \(\mathbb{G}_a\) (char \(p\)).
Role in reductive structure
For any connected affine algebraic group \(H\) over \(\bar k\):
- The unipotent radical \(R_u(H)\) is the largest connected unipotent normal subgroup.
- \(H / R_u(H)\) is reductive.
- For \(H\) reductive, \(R_u(H) = 1\) — this is the definition.
- For \(H\) a parabolic subgroup of a reductive \(G\): \(R_u(H)\) is the "unipotent radical of the parabolic," and the Levi \(H / R_u(H)\) is reductive.
Role in representation theory
- Whittaker models. Generic characters of the unipotent radical of a Borel give Whittaker functionals on \(G(F)\)-representations.
- Bernstein-Zelevinsky. Derivatives along unipotent radicals of parabolics are core invariants of \(\mathrm{GL}_n\)-representations.
- Nilpotent orbits. \(\operatorname{Lie}(U)\) for the unipotent radical of a parabolic captures nilpotent orbits via the Bala-Carter classification.
Mathlib
Not yet formalised. Examples like \(\mathbb{G}_a\) exist via the project's basic examples node.