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Unipotent groups are exactly the subgroups of some U_n, equivalently the groups with coconnected coordinate Hopf algebra

Statement

Let $k$ be a field and let $G$ be an affine algebraic group over $k$ (an affine group scheme of finite type over $k$ , Affine schemes and their coordinate rings, Group schemes of finite type over a field). Consider the following conditions:

(a) $G$ is unipotent (Unipotent algebraic groups and unipotent representations);

(b) $G$ is isomorphic to a closed subgroup scheme of the upper unitriangular group scheme $U_{n}$ for some $n \geq 1$ (The upper unitriangular group scheme U_n and its coordinate ring);

Without a choice assumption, (c) implies (a); more generally a surjective Hopf-algebra quotient of $O (U_{n})$ is coconnected and therefore defines a unipotent group. Assuming the Axiom of Choice (The Axiom of Choice), (a) implies (b), and the geometric closed-subgroup conversion in (b) implies (c), so all three conditions are equivalent; the same assumption makes them equivalent to existence of a faithful finite-dimensional unipotent rational representation. The exact uses of AC are the faithful-representation/closed-immersion suppliers constructing the triangular embedding in (a) implies (b), and the closed-subgroup/quotient-ring supplier in geometric (b) implies (c). The Hopf-quotient filtration argument and (c) implies (a) use no choice. No smoothness, connectedness or perfectness is assumed; in positive characteristic the infinitesimal group $α_{p}$ and the constant group $(Z / p Z)_{k}$ are unipotent groups that are non-smooth, respectively non-connected.

Facts & Assumptions

Given: A field $k$ and an affine algebraic group $G$ over $k$ ; AC is assumed only for geometric (a) implies (b), geometric (b) implies (c), and the faithful-existence reformulation.

[F1]

$G$ is unipotent when every nonzero rational representation of $G$ has a nonzero fixed vector, equivalently every simple rational representation is one-dimensional with trivial action; it suffices to test finite-dimensional representations, and every finite-dimensional representation of a unipotent group is unipotent. (Unipotent algebraic groups and unipotent representations, Unipotence is equivalent to unipotence of all finite-dimensional representations)

[F2]

Assume AC. For every affine finite-type group scheme over $k$ there is a faithful finite-dimensional rational representation, i.e. a monomorphism $G \to {G L}_{n}$ ; every monomorphism of finite-type group schemes over a field is a closed immersion. (Affine finite-type group schemes have faithful finite-dimensional representations, Finite-type algebraic group monomorphisms are closed immersions)

[F3]

$U_{n}$ is the closed subgroup scheme of ${G L}_{n}$ of upper unitriangular matrices; its coordinate ring $O (U_{n}) = k [X_{i j} : i < j]$ is coconnected, A surjective Hopf-algebra quotient of $O (U_{n})$ is coconnected, without choice. Assuming AC, the coordinate ring of a geometric closed subgroup scheme of $U_{n}$ is such a quotient. (The upper unitriangular group scheme U_n and its coordinate ring, Coconnected Hopf algebras: the coordinate ring of U_n and passage to quotients, Closed subgroup schemes of an affine group scheme correspond to Hopf ideals)

[F4]

If $A$ is a coconnected commutative Hopf algebra over $k$ and $V \neq 0$ is an $A$ -comodule, then $V$ has a nonzero vector fixed by the comodule structure; for $A = O (G)$ this says that every nonzero rational representation of $G$ has a nonzero $G$ -fixed vector. (Coconnected Hopf algebras give fixed vectors in every nonzero comodule)

Proof

Given: A field $k$ and an affine algebraic group $G$ over $k$ ; AC is assumed only for geometric (a) implies (b), geometric (b) implies (c), and the faithful-existence reformulation.

1.1F1F2

Assume (a) and the Axiom of Choice. By [F2] choose a faithful finite-dimensional representation $G ↪ G L (V)$ , adjoining a trivial line if needed to ensure $\dim V \geq 1$ . Since $G$ is unipotent, [F1] makes this representation unipotent, so there is a basis of $V$ in which every element of $G$ acts by an upper unitriangular matrix; therefore the closed immersion $G ↪ G L (V)$ factors through the closed subgroup scheme $U_{n}$ of ${G L}_{n}$ for $n = \dim V$ . Hence (b) holds.

1.2F3

Assume (b) and AC for the geometric quotient-ring conversion: $G$ is a closed subgroup scheme of some $U_{n}$ . By [F3] the coordinate ring of a closed subgroup scheme of $U_{n}$ is a quotient of the coconnected Hopf algebra $O (U_{n})$ by a Hopf ideal, hence coconnected. Hence (c) holds.

1.3F1F4

Assume (c). Let $V \neq 0$ be a nonzero rational representation of $G$ . Since $O (G)$ is coconnected, [F4] provides a nonzero vector $v \in V$ with $v$ fixed by $G$ . Thus every nonzero rational representation of $G$ has a nonzero fixed vector, so $G$ is unipotent by [F1]. Hence (a) holds.

2.1step 1.1step 1.2step 1.3∎

Step 1.3 gives the choice-free implication (c) implies (a), and the algebraic Hopf-quotient filtration in [F3] is also choice-free. Under AC, steps 1.1 and 1.2 give (a) implies (b) implies (c), yielding the full equivalence. Under that same assumption, for the final reformulation: a faithful unipotent finite-dimensional representation produces a closed immersion into some $U_{n}$ by the argument of [step 1.1], and conversely a closed immersion $G ↪ U_{n}$ followed by the inclusion $U_{n} \subseteq {G L}_{n}$ is a faithful unipotent representation; so the existence of such a representation is equivalent to (b).

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Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
Florian Herzig, Linear Algebraic Groups (University of Toronto lecture notes, 2013) (standard reference, not scraped)