Unipotent and Solvable Groups and Borel Fixed Points

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

(a) Unipotent representations. A finite-dimensional rational representation $(V,r,\rho )$ of $G$ (Rational representations and comodules of an affine group scheme, Vector space over a field) is unipotent if there is a $k$-basis of $V$ (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis) such that, for every commutative $k$-algebra $R$ and every $g\in G(R)$, the operator $r_R(g)$ on $V{\otimes }_{k}R$ is upper triangular with all diagonal entries equal to $1$ in the scalar-extended basis. Equivalently, by the comodule dictionary, $V$ is unipotent if and only if $V$ has a complete $G$-stable flag $V=V_m\supseteq V_{m-1}\supseteq \cdots \supseteq V_0=0$ with $G$ acting trivially on each quotient $V_i\mathrm{/}{V}_{i-1}$; the equivalence uses that a complete flag with trivial successive quotients is exactly a chain of subspaces in a basis as above, and conversely.

(b) Unipotent groups. The group $G$ is unipotent if every nonzero rational representation of $G$ has a nonzero $G$-fixed vector, equivalently if every simple rational representation of $G$ is one-dimensional with trivial action. Because every rational representation is a union of finite-dimensional subrepresentations (Every element of a comodule lies in a finite-dimensional subcomodule), it suffices to test finite-dimensional representations: $G$ is unipotent if and only if every nonzero finite-dimensional rational representation has a nonzero fixed vector.

No smoothness, reducedness or connectedness of $G$ is imposed; the matrix formulation in (a) is written out because the group scheme $U_n$ of upper unitriangular matrices is introduced separately and is not used in the definition.

Let $k$ be a field and let $(A,\mathrm{\Delta },\epsilon ,S)$ be a commutative Hopf algebra over $k$ (Commutative Hopf algebras over a field), with tensor products over $k$ (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums).

The Hopf algebra $A$ is coconnected if there is an increasing filtration $C_0\subseteq C_1\subseteq C_2\subseteq \cdots$ of $A$ by $k$-linear subspaces (Linear subspace of a vector space) with $C_0=k\cdot 1_A,{\bigcup }_{r\ge 0}{C}_r=A,\mathrm{\Delta }(C_r)\subseteq {\sum }_{i=0}^r{C}_i{\otimes }_k{C}_{r-i}\text{for all }r\ge 0.$ In words: the filtration starts at the constants, exhausts $A$, and the comultiplication of an element of filtration degree $r$ lands in the sum of tensor products whose degrees add up to $r$.

No reducedness, finite generation or smoothness of $A$ is imposed, and no choice principle is used. The filtration is the condition dual to the existence of a group-like element generating the simple comodules; the first step is the constants because a unipotent group has no nontrivial characters.

Let $k$ be a field and let $n\ge 1$. Let ${\mathrm{G}\mathrm{L}}_n$ be the general linear group scheme over $k$ with its coordinate ring $k[x_{ij},{\det }^{-1}]$ (The general linear group scheme and its coordinate ring) and let $T_n$, $D_n$, $U_n$ be the closed subschemes of ${\mathrm{G}\mathrm{L}}_n$ defined by the following equations on the matrix entries, cut out as quotients of the coordinate ring of ${\mathrm{G}\mathrm{L}}_n$ (Closed immersions into affine schemes are quotient spectra):

• $T_n$: the upper triangular matrices, defined by $x_{ij}=0$ for $i>j$;
• $D_n$: the diagonal matrices, defined by $x_{ij}=0$ for $i\mathrm{\ne }j$;
• $U_n$: the upper unitriangular matrices, defined by $x_{ij}=0$ for $i>j$ and $x_{ii}=1$.

These are closed subgroup schemes of ${\mathrm{G}\mathrm{L}}_n$ (Morphisms and closed subgroup schemes of group schemes), because for each of them the defining equations are stable under matrix multiplication, inverse and identity; equivalently, by the valued-point criterion (Closed subgroup schemes are detected on all algebra-valued points), for every commutative unital $k$-algebra $R$ the $R$-points are the corresponding subgroups $T_n(R),D_n(R),U_n(R)$ of ${\mathrm{G}\mathrm{L}}_n(R)$, described by the same equations. In particular $U_n(R)=\{({\alpha }_{ij})\in {\mathrm{G}\mathrm{L}}_n(R):{\alpha }_{ij}=0(i>j),{\alpha }_{ii}=1\},$ the group of upper unitriangular matrices, and $T_n=D_n⋉U_n$ is the group of invertible upper triangular matrices, the semidirect product for the conjugation action of $D_n$ on $U_n$ (Upper triangular, lower triangular and diagonal square matrices over a commutative ring).

The coordinate ring of $U_n$ is $O(U_n)=k[X_{ij}\mid 1\le i<j\le n]$ with comultiplication $\mathrm{\Delta }(X_{ij})=X_{ij}\otimes 1+1\otimes X_{ij}+{\sum }_{i<l<j}{X}_{il}\otimes X_{lj},$ counit $\epsilon (X_{ij})=0$ and antipode determined by the inverse of a unitriangular matrix; the displayed formula is the matrix multiplication formula restricted to unitriangular matrices and visibly preserves the polynomial ring, so $O(U_n)$ is a polynomial algebra on the $\left(\genfrac{}{}{0px}{}{n}{2}\right)$ entries strictly above the diagonal (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis).

Let $k$ be a field and let $G$ be an algebraic group over $k$, that is, a group scheme of finite type over $k$ (Group schemes of finite type over a field), with multiplication $m$, inverse $i$ and identity section $e$.

The commutator morphism is $c:G{\times }_{k}G\to G,c(g,h)=gh{g}^{-1}{h}^{-1}\mathrm{.}$ It is a $k$-morphism: the map $(g,h)\mapsto (g,h,g^{-1},h^{-1})$ is built from $i$ and the universal property of the fibre product (Fibre product of schemes), and $c$ is obtained from it by iterated multiplication. For every $k$-algebra $R$, the induced map $G(R)\times G(R)\to G(R)$ is the group commutator.

The derived subgroup $DG$ is the smallest closed subgroup scheme $H\subseteq G$ (Morphisms and closed subgroup schemes of group schemes) through which $c$ factors, that is, such that $c$ equals the composite of a morphism $G{\times }_{k}G\to H$ with the inclusion $H\hookrightarrow G$. Equivalently, $DG$ is the smallest closed subgroup scheme of $G$ containing the scheme-theoretic image of $c$ (Scheme-theoretic image, Scheme-theoretic image of a quasi-compact morphism). It exists: if $G$ is affine, closed subgroup schemes correspond to Hopf ideals of $O(G)$ and $c$ corresponds to a $k$-algebra map ${\mathrm{\Delta }}_c:O(G)\to O(G){\otimes }_{k}O(G)$, and the Hopf ideals contained in $\ker {\mathrm{\Delta }}_c$ are closed under sums, so their largest element cuts out $DG$; in general the closed subgroup schemes through which $c$ factors are closed under schematic intersection, their intersection is computed by the ideal sheaf generated by the defining ideal sheaves, and $G$ is Noetherian because it is of finite type over a field.

The derived series of $G$ is the sequence $D^{0}G=G,D^{i+1}G=D(D^{i}G)(i\ge 0),$ so each $D^{i}G$ is a closed subgroup scheme of $G$ for $i\ge 1$. The group $G$ is solvable if $D^{n}G=1$ for some $n\ge 0$, where $1$ denotes the trivial subgroup scheme, the image of the identity section $e$.

A closed subgroup scheme $N\subseteq G$ is normal if it is stable under the conjugation action of $G$ on itself: the morphism $G{\times }_{k}N\to G$, $(g,n)\mapsto gn{g}^{-1}$, factors through the inclusion $N\hookrightarrow G$. For $k$-algebras $R$ this says exactly that $N(R)$ is a normal subgroup of $G(R)$. This is the sense of normality used throughout this page.

Let $k$ be a field. An affine algebraic group $G$ 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) is trigonalizable if every simple rational representation of $G$ (Rational representations and comodules of an affine group scheme) has dimension $1$ over $k$.

Equivalently, by the criterion proved as Trigonalizable groups, invariant flags and embeddings into T_n, every finite-dimensional rational representation of $G$ admits a basis in which $G$ acts through upper triangular matrices, i.e. the representation is isomorphic to one factoring through the upper triangular group scheme $T_n$ of some ${\mathrm{G}\mathrm{L}}_n$.

Both unipotent groups (Unipotent algebraic groups and unipotent representations) and diagonalizable groups are trigonalizable: for a unipotent group every simple representation is trivial of dimension one by definition, and for a diagonalizable group the character eigenspace decomposition exhibits every simple representation as one-dimensional. The formulation by simple representations is the one used in the induction proving Lie-Kolchin; the flag formulation is the one used to embed trigonalizable groups into $T_n$.

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

A torus of $G$ is a closed subgroup scheme $T\subseteq G$ (Morphisms and closed subgroup schemes of group schemes) whose base extension to a separable closure of $k$ is isomorphic to ${\mathbf{G}}_m^r$ for some integer $r\ge 0$; it is a split torus if that isomorphism is already defined over $k$ (Groups of multiplicative type and tori). It is a maximal torus if it is maximal with respect to inclusion among the tori of $G$. Equivalently, $T$ is a closed subgroup that is geometrically a product of copies of ${\mathbf{G}}_m$, and no strictly larger torus of $G$ contains it; for smooth connected affine groups, maximality is preserved by every field extension (Conrad, Grothendieck’s theorem on tori, Corollary 1.3, printed p. 1).

A Borel subgroup of a smooth $G$ is a smooth connected solvable closed subgroup scheme $B\subseteq G$ whose base extension to an algebraic closure is maximal among smooth connected solvable closed subgroup schemes. Over an algebraically closed field this means exactly that $B$ is a maximal connected solvable subgroup variety, with its reduced smooth structure (The derived subgroup, the derived series and solvable algebraic groups, Smooth morphism of schemes). Maximality here is among subgroup varieties, not arbitrary possibly infinitesimal subgroup schemes. A Borel subgroup need not be defined over a general field; existence and conjugacy are asserted below over an algebraically closed field.

A Borel pair is a pair $(B,T)$ consisting of a Borel subgroup $B$ and a maximal torus $T$ with $T\subseteq B$.

On this page Borel subgroups are used only for smooth $G$ (Smooth morphism of schemes); Borel subgroups are smooth by the subgroup-variety convention, and the existence and conjugacy theorems are proved later on this page. Maximal tori exist whenever $G$ has a torus, by maximizing dimension among tori, since a strict inclusion of tori increases dimension; the existence of a Borel subgroup containing a given torus is proved where it is used.

Let $k$ be a field, let $G$ be an affine algebraic group over $k$ (Affine schemes and their coordinate rings, Group schemes of finite type over a field) and let $M$ be a commutative affine algebraic group over $k$ on which $G$ acts by group automorphisms, $\cdot :G{\times }_{k}M\to M$ (Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers). Form the semidirect product $M⋊G$, the $k$-group scheme whose $R$-points are the pairs $(m,g)\in M(R)\times G(R)$ with multiplication $(m,g)(m^{\mathrm{\prime }},g^{\mathrm{\prime }})=(m+g\cdot m^{\mathrm{\prime }},g{g}^{\mathrm{\prime }});$ the product is a group law because the action is by group automorphisms, and $M⋊G$ is constructed from the given morphisms using fibre products (Fibre product of schemes).

A crossed homomorphism is a morphism of $k$-schemes $f:G\to M$ such that $f(xy)=f(x)+x\cdot f(y)$ for all $x,y\in G(R)$ and all $k$-algebras $R$. It is principal if there is an element $m\in M(k)$ with $f(x)=x\cdot m-m\text{for all }x\in G(R),R\mathrm{.}$

The assignments $x\mapsto (f(x),x)$ and $(m,x)\mapsto x$ identify sections $G\to M⋊G$ of the projection $M⋊G\to G$ with crossed homomorphisms: indeed $(f(x),x)(f(y),y)=(f(x)+x\cdot f(y),xy)$, so multiplicativity of the section is exactly the displayed identity. Conjugating a section by $m\in M(k)$ changes the corresponding crossed homomorphism by the principal crossed homomorphism $x\mapsto x\cdot m-m$; hence two sections are conjugate by an element of $M(k)$ if and only if their crossed homomorphisms differ by a principal one.

An extension of group functors $0\to M\to E\to G\to 1$ is a sequence of group functors on $k$-algebras that is exact, with $E\to G$ the given projection. Such an extension is a Hochschild extension if the projection $E\to G$ admits a section as a map of set-valued functors. For a Hochschild extension the conjugation action of $G$ on $M$ induced by any such section is independent of the choice of section, and the equivalence classes of Hochschild extensions inducing a given action are classified by the second Hochschild cohomology group of Hochschild cohomology of algebraic groups and the classification of Hochschild extensions.

Let $k$ be a field and let $G$ be an affine algebraic group over $k$ (Affine schemes and their coordinate rings, Group schemes of finite type over a field). Then $G$ is unipotent (Unipotent algebraic groups and unipotent representations) if and only if every finite-dimensional rational representation of $G$ (Rational representations and comodules of an affine group scheme) is unipotent. In particular, if $G$ is unipotent then every nonzero finite-dimensional rational representation admits a basis in which $G$ acts through the upper unitriangular group scheme $U_n$ of The upper unitriangular group scheme U_n and its coordinate ring, and the class of unipotent finite-dimensional representations is closed under subquotients, direct sums and tensor products.

Let $k$ be a field and $n\ge 1$. Then:

(a) $O(U_n)=k[X_{ij}\mid 1\le i<j\le n]$ is a coconnected Hopf algebra: assigning weight $j-i$ to each generator $X_{ij}$ and letting $C_r$ be the span of the monomials of weight at most $r$ gives a filtration satisfying $\mathrm{\Delta }(C_r)\subseteq {\sum }_{i+j=r}{C}_i{\otimes }_k{C}_j$;

(b) if $A\to B$ is a surjective morphism of commutative Hopf algebras over $k$ and $A$ is coconnected, then $B$ is coconnected;

(c) assuming the Axiom of Choice (The Axiom of Choice), if $H\subseteq U_n$ is a closed subgroup scheme, then $O(H)$ is a quotient of $O(U_n)$ and hence coconnected.

Let $A$ be a coconnected commutative Hopf algebra over a field $k$, with filtration $C_0\subseteq C_1\subseteq \dots$ as in Coconnected commutative Hopf algebras. If $V\mathrm{\ne }0$ is an $A$-comodule with coaction $\rho :V\to V{\otimes }_{k}A$, then $V$ has a nonzero vector $v$ with $\rho (v)=v\otimes 1$.

In particular, if $G$ is an affine algebraic group over $k$ whose coordinate Hopf algebra $O(G)$ is coconnected, then every nonzero rational representation of $G$ has a nonzero fixed vector; that is, $G$ is unipotent in the sense of Unipotent algebraic groups and unipotent representations.

Let $k$ be a field and $n\ge 1$, and let $U_n\subseteq T_n\subseteq {\mathrm{G}\mathrm{L}}_n$ be the upper unitriangular and upper triangular group schemes of The upper unitriangular group scheme U_n and its coordinate ring, with $T_n=D_n⋉U_n$.

Order the pairs $(i,j)$ with $1\le i<j\le n$ by increasing $j-i$ (and arbitrarily, say by increasing $i$, within a fixed difference), and let $m=n(n-1)\mathrm{/}2$. For $0\le r\le m$ let $U_n^{(r)}$ be the closed subgroup scheme of $U_n$ of matrices whose entries $x_{ij}$ vanish on the first $r$ pairs of the ordering; thus $U_n^{(0)}=U_n$ and $U_n^{(m)}=1$, and the $U_n^{(r)}$ are closed subgroup schemes of $U_n$ stable under conjugation by $T_n$.

Then $U_n=U_n^{(0)}\supseteq U_n^{(1)}\supseteq \cdots \supseteq U_n^{(m)}=1$ is a central series of closed subgroup schemes of $U_n$ stable under $T_n$: $[U_n^{(r)},U_n]\subseteq U_n^{(r+1)}$ for $0\le r<m$, and each successive quotient $U_n^{(r)}\mathrm{/}{U}_n^{(r+1)}$ is canonically isomorphic to ${\mathbf{G}}_a$, the isomorphism being given by the coordinate $x_{ij}$ of the $(r+1)$-st pair. The diagonal torus $D_n$ acts on each quotient through the character $d\mapsto d_i{d}_j^{-1}$.

Assume the Axiom of Choice for the faithful triangular embedding and the affine kernel/image quotient suppliers (The Axiom of Choice).

Let $k$ be a field and let $G$ be an affine unipotent algebraic group over $k$ (Affine schemes and their coordinate rings, Group schemes of finite type over a field). Then there is a central series $G=G_0\supseteq G_1\supseteq \cdots \supseteq G_r=1$ of closed subgroup schemes of $G$ whose successive quotients are isomorphic to closed subgroup schemes of ${\mathbf{G}}_a$. In particular every unipotent algebraic group is nilpotent and hence solvable.

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 $n\ge 1$ (The upper unitriangular group scheme U_n and its coordinate ring);

(c) the coordinate Hopf algebra $O(G)$ is coconnected (Coconnected commutative Hopf algebras).

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 ${\alpha }_p$ and the constant group $(\mathbb{Z}\mathrm{/}p\mathbb{Z}{)}_k$ are unipotent groups that are non-smooth, respectively non-connected.