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An abelian variety over a field is a complete, geometrically connected group variety; equivalently it is a connected projective algebraic group in the usual smooth variety setting. Its group operation is necessarily commutative, a consequence of rigidity rather than an extra condition hidden in the name. An elliptic curve with a chosen origin is a one-dimensional abelian variety; products of elliptic curves give higher-dimensional examples. The additive and multiplicative algebraic groups are not complete and hence are not abelian varieties. Their geometry links line bundles, torsion elements and the Jacobian variety of a curve to questions in number theory.^[ref]

An algebraic torus over a field $k$ is an algebraic group $T$ whose base change to a separable closure $k_s$ is isomorphic to ${\mathbb{G}}_m^n$. Its dimension is $n$. An algebraic torus already isomorphic to ${\mathbb{G}}_m^n$ over $k$ is a split algebraic torus. Toric geometry described by an ordinary toric fan uses a split algebraic torus.

The real algebraic group $x^2+y^2=1$, with multiplication inherited from complex multiplication, is a one-dimensional nonsplit example. Over the complex numbers, $z=x+iy$ identifies its coordinate ring with $\mathbb{C}[z,z^{-1}]$. Complex conjugation acts on its geometric character lattice of an algebraic torus by $m\mapsto -m$, so it has no nontrivial real-defined algebraic torus character. Algebraic tori of dimension one therefore need not be the real multiplicative group.^[ref]

Let $S$ be constructible in an algebraic group, and $C=\overline{S}$. The closure is a subgroup. Choose a dense open $U\subseteq C$ contained in $S$. For every $c\in C$, the dense opens $U,cU$ intersect, giving $u=cv$ for $u,v\in S$. Hence $c=u{v}^{-1}\in S$ and $S=C$. The Chevalley constructibility theorem therefore proves closedness of homomorphism images.

For an algebraic group defined over ${\mathbb{F}}_q$, its Frobenius morphism acts as entrywise $q$th powers in a standard matrix realization. Its differential is zero, and its fixed-point group is finite. A conjugate $F^{\prime }=\phi F{\phi }^{-1}$ defines an isomorphic rational structure. This group morphism must be distinguished from a field Frobenius endomorphism alone and from the convention called geometric Frobenius in Galois theory, which is inverse to the arithmetic generator on field points.

For an algebraic group $G$ acting on a algebraic variety $X$ over an algebraically closed field, a geometric quotient is a surjective invariant morphism $q:X\to Y$ whose fibers are exactly the group orbits, whose topology is the quotient topology, and whose regular functions are the invariant regular functions locally:The orbit condition gives actual orbit parameters; the function-sheaf condition specifies the algebraic structure on the orbit space. Both matter beyond a set-theoretic quotient.

For scalar multiplication on ${\mathbb{A}}^{n+1}\setminus \{0\}$, the map $x\mapsto [x]$ is the geometric quotient by ${\mathbb{G}}_m$, giving projective space. On a standard chart, coordinate ratios are the invariant functions. In the Cox construction, the quotient is geometric for a simplicial toric fan; in general the categorical quotient can identify more than a single orbit.^[ref]

A finite group of Lie type is built from an algebraic group in positive characteristic by taking fixed points of a suitable Frobenius-type endomorphism, often followed by a derived-group or central-quotient construction. The terminology covers classical matrix families and exceptional families, including twisted forms. It connects the structure of Lie theory with finite group theory; such a finite group is not a positive-dimensional real Lie group.^[ref]

For example, ${\mathrm{SL}}_2({\mathbb{F}}_q)$ consists of determinant-one $2\times 2$ matrices over a finite field. There are $q^2-1$ choices for its first column; for each nonzero first column, the determinant-one equation has $q$ solutions for the second. Dividing by its scalar center gives the projective special linear groupFor $q=5$, its order is sixty. These projective groups are simple for $q>3$, whereas ${\mathrm{PSL}}_2(2)\cong S_3$ and ${\mathrm{PSL}}_2(3)\cong A_4$ are exceptions.^[ref]

The classification of finite simple groups uses the simple members of the Lie-type families, with exceptional small parameters treated explicitly. A family name alone does not guarantee simplicity.

An invariant differential on a group scheme is a differential one-form preserved by every left translation. For a smooth group scheme $G$ over a field $k$, translation identifiesThus a covector at the identity extends uniquely to a left-invariant form. On the additive algebraic group the invariant form is $dx$; on the multiplicative algebraic group it is $dx/x$, since $d(ax)/(ax)=dx/x$. An abelian variety of dimension $g$ therefore has $g$ linearly independent invariant one-forms, providing the local linear data behind its global group geometry.

An isogeny is a surjective morphism of algebraic groups with finite kernel. For abelian varieties it is a finite surjective group morphism. An isogeny of elliptic curves is its one-dimensional abelian-variety specialization.

James E. Humphreys wrote influential mathematical expositions on Lie theory and representation theory. His book Modular Representations of Finite Groups of Lie Type by James E. Humphreys develops modular representation theory for finite groups of Lie type, relating finite groups to algebraic groups over finite fields. The opening chapter describes constructions using rational points and fixed points of appropriate endomorphisms.^[ref]

A small example illustrates why the characteristic of a field matters in representation theory. Over the finite field ${\mathbb{F}}_2$, the matrixsatisfies $A^2=I$ but $A\ne I$, so it represents the generator of a group of order two. Its fixed vectors form only the line spanned by $(1,0)$. That line has no invariant one-dimensional complement: over ${\mathbb{F}}_2$, an invertible action on a line must be the identity, so two invariant lines would force $A=I$. The example shows concretely how a representation can fail to split into smaller pieces when the field characteristic divides the group order. This is part of the mathematical setting that Humphreys's expositions make accessible.

For a connected smooth algebraic group defined over a finite field, work over its algebraic closure and let $F$ be the finite-field Frobenius. Then the Lang map $g\mapsto g^{-1}F(g)$ is surjective. One proof uses the twisted action $x\cdot y=xyF(x{)}^{-1}$. Since $dF=0$, every orbit map has surjective differential, so each orbit is open; connectedness leaves only one orbit. The same differential calculation makes the Lang map étale, with finite fibres equal to left cosets of $G^F$.

A maximal algebraic torus is an algebraic torus not properly contained in another algebraic torus of the ambient algebraic group. Over an algebraically closed field, maximal algebraic tori in a connected reductive algebraic group are conjugate. A Borel subgroup contains one, and it splits as $B=T⋉R_u(B)$ with its unipotent radical.

For $G{L}_n$, the invertible diagonal matrices form a maximal algebraic torus isomorphic to ${\mathbb{G}}_m^n$. Choosing a different basis conjugates this example. The dimension of a maximal algebraic torus is the absolute rank of the connected reductive group.

Over a general field, geometric conjugacy does not assert conjugacy by rational points. In $S{L}_2(\mathbb{R})$, the split diagonal algebraic torus has noncompact real points, while the rotation norm-one algebraic torus has compact real points, so they cannot be conjugate by a real matrix. Both become maximal split algebraic tori after passing to the complex numbers.^[ref]

The multiplicative algebraic group has ${\mathbb{G}}_m(K)=K^{\times }$, with multiplication as its group operation. Its coordinate ring is $K[z,z^{-1}]$. The formal multiplicative group describes its law near the identity $z=1$.

Write $k={\overline{\mathbb{F}}}_q$, and let $F_0$ be the entrywise $q$th-power Frobenius endomorphism of an algebraic group on the general linear group $G={\mathrm{GL}}_2(k)$. The algebraic closure is visible in the original PDF; it is lost in the converted TeX. Let $T_0$ be the diagonal maximal algebraic torus and $B_0$ the upper triangular Borel subgroup.

Choose $\beta \in k$ with $\beta -{\beta }^q=1$. Such a $\beta$ exists because $k$ is an algebraic closure, and $\beta \notin {\mathbb{F}}_q$. PutTransporting the standard ${\mathbb{F}}_q$-model through the algebraic group automorphism $\mathrm{Int}(h_{\pm })$ gives a rational structure on an algebraic group over the same field ${\mathbb{F}}_q$. In particular these are genuine Frobenius endomorphisms of an algebraic group, and their fixed-point groups areThe auxiliary $\beta$ need not be rational for the original structure: it changes the identification with the geometric group, not the field over which the transported model is defined.

For either sign, take the rational maximal torus and Borel subgroupIndeed $F_{\pm }(h_{\pm }H{h}_{\pm }^{-1})=h_{\pm }{F}_0(H)h_{\pm }^{-1}$ for $H=T_0,B_0$, so both subgroups are $F_{\pm }$-stable. Conjugacy also preserves maximality of the algebraic torus and the property of being a Borel subgroup.

Finally,Neither $c_{+}$ nor $c_{-}$ is central, so neither endomorphism is $F_0$. If $F_{+}=F_{-}$, surjectivity of $F_0$ on $G(k)$ would imply that $c_{-}^{-1}{c}_{+}$ is central. Butis not scalar in any characteristic. Thus the two nonstandard rational structures have distinct Frobenius endomorphisms, including when $q=2$.

The dimension of a topological space by irreducible chains is the supremum of the integers $d$ for which there is a chain $Z_0⊊Z_1⊊\cdots ⊊Z_d$ of nonempty irreducible closed subsets of the space. Closedness here is relative to the given locally closed space. For varieties this is their Krull dimension.

An algebraic group is a group whose underlying space is an algebraic variety and whose multiplication $G\times G\to G$ and inversion $G\to G$ are morphisms. An algebraic group action on a variety $V$ is a morphism $G\times V\to V$ satisfying the identity and associativity axioms of a group action.

For the homomorphism $\phi$, the kernel is ${\phi }^{-1}(1_H)$, so the kernel is closed. To prove closedness of the image, use the Chevalley constructibility theorem: the image of a morphism of varieties is constructible. Thus $S=\phi (G)$ is a constructible subset of a variety and an abstract subgroup. Its closure $C$ is also a subgroup: translation by elements of $S$ preserves $C$, and continuity then extends multiplication and inversion to $C$.

A dense constructible subset contains a dense open subset $U$ of its closure. For $c\in C$, both $U$ and $cU$ are dense open subsets of $C$, so their intersection is nonempty. If $u=cv$ with $u,v\in U\subseteq S$, then $c=u{v}^{-1}\in S$. Hence $S=C$, proving that the image is closed. This is the principle that a constructible subgroup is closed.

Every nonempty fiber of $G\to S$ is a translate of $\ker \phi$ and has that same dimension. The fiber dimension theorem therefore givesThis dimension formula for an algebraic group homomorphism is a dimension statement, so it does not require separability of $\phi$.

For a dimension vector of a quiver representation $\mathbf{n}$, setThe base change action on quiver representations isThe entries are regular functions on the product of the general linear groups and the quiver representation space, because inverse entries are cofactors divided by the invertible determinant. Thus this is an algebraic group action.

For nonzero $\mathbf{n}$, let $\Delta k^{\times }$ be the common scalar subgroup $(\lambda I_{n_i}{)}_i$ and define $\mathrm{PGL}(\mathbf{n})=\mathrm{GL}(\mathbf{n})/\Delta k^{\times }$. Common scalars act trivially, so the formula descends to an algebraic group action of this projective base change group of a quiver. This is a quotient by one common scalar, not a product of the individual projective groups. If every $n_i=0$, the representation space is a point and both actions are taken to be trivial.

An algebraic group is connected when its underlying Zariski topology is a connected space. An unipotent algebraic group is a linear algebraic group whose elements, in a faithful matrix realization, satisfy that $u-I$ is nilpotent. A reductive algebraic group, in the connected convention, is a connected linear algebraic group whose unipotent radical is trivial: it has no nontrivial connected normal unipotent algebraic subgroup.

Let $E={\mathrm{End}}_Q(X)$. It is a finite-dimensional associative algebra, and ${\mathrm{Aut}}_Q(X)=E^{\times }$. The latter is the nonempty open subset where the determinant on ${⨁}_i{X}_i$ is nonzero. Since the affine space $E$ is irreducible, this open subset is irreducible and hence connected. It is a linear algebraic group: within $\mathrm{GL}({⨁}_i{X}_i)$ it is cut out by the linear equations of preserving the vertex summands and commuting with the arrow maps. Thus the automorphism group is connected.

By the Krull–Schmidt theorem, write $X\cong {⨁}_{a=1}^s{M}_a^{\oplus m_a}$, with pairwise nonisomorphic indecomposables $M_a$. The Fitting lemma makes each ${\mathrm{End}}_Q(M_a)$ a local algebra. Its residue division algebra is $k$: every element of a finite-dimensional division algebra over the algebraically closed field has an eigenvalue for left multiplication, and subtracting that scalar gives a noninvertible element, hence zero.

The standard radical description of the endomorphism algebra of a Krull-Schmidt decomposition therefore givesConcretely, maps between nonisomorphic summands lie in the radical, while on each isotypic block one reduces all entries modulo the local radical. The off-diagonal rule follows because a composition $M_a\to M_b\to M_a$ cannot be a unit for $a\ne b$: it would split $M_a$ off the indecomposable $M_b$, forcing an isomorphism. The finite-dimensional radical description then identifies the kernel of this block reduction with $J(E)$. Put $U=1+J(E)$. The Jacobson radical of the finite-dimensional algebra is nilpotent, so every $1+j$ is a unit and is unipotent on ${⨁}_i{X}_i$. Also $U$ is closed, being the translate of the linear subspace $J(E)$, and normal, being the kernel ofThis map has an explicit algebraic section: after fixing the direct-sum decomposition, a matrix $B_a$ acts on the $m_a$ copies by $B_a\otimes 1_{M_a}$. These block actions give a subgroup $L\cong {\prod }_a{\mathrm{GL}}_{m_a}(k)$ with trivial intersection with $U$. Every unit is uniquely a product of an element of $U$ and one of $L$. HenceThis Levi decomposition of a quiver automorphism group holds in every characteristic; the section is constructed directly from the multiplicity spaces.

A rational structure over ${\mathbb{F}}_q$ is a model over that finite field together with its identification after extension to ${\overline{\mathbb{F}}}_q$. It determines a Frobenius endomorphism of an algebraic group, whose fixed points are the rational points. Conjugating a standard Frobenius by an algebraic group automorphism transports the rational structure, even if the automorphism is not itself rational for the original structure.

For $q=3^{2a+1}\ge 27$, the small Ree group ${}^2{G}_2(q)$ is a twisted finite group of Lie type. In characteristic three, the algebraic group of type G2 has an exceptional endomorphism $\sigma$ with ${\sigma }^2=F_3$, the Frobenius endomorphism of an algebraic group. The fixed points of ${\sigma }^{2a+1}$ form ${}^2{G}_2(q)$. This explains both the twist and the odd exponent restriction.^[ref]

It is simple, has order $q^3(q^3+1)(q-1)$, and acts two-transitively on $q^3+1$ points. At $q=27$ the degree is $19684$ and the order is $10073444472$. The excluded parameter $q=3$ gives a nonsimple group. The characteristic-two family ${}^2{F}_4(q)$ is a different Ree family.

A split algebraic torus over a field $k$ is an algebraic group isomorphic over $k$ to a product ${\mathbb{G}}_m^r$ of multiplicative group schemes. Its coordinate ring is the Laurent polynomial ringand its $k$-points are $(k^{\times }{)}^r$. Choosing such an isomorphism supplies torus coordinates; the torus need not come with preferred coordinates.

Every algebraic torus character is a monomial $x_1^{m_1}\cdots x_r^{m_r}$ with integer exponents. The character group is therefore ${\mathbb{Z}}^r$, and cocharacters form the dual lattice. This integral structure makes split tori the dense groups in ordinary fan-based toric geometry.

The adjective split concerns the field of definition, not connectedness of the real points. For example, ${\mathbb{G}}_m$ is split over $\mathbb{R}$ although ${\mathbb{R}}^{\times }$ has two connected components. The real norm-one algebraic torus from $\mathbb{C}/\mathbb{R}$ is connected as a real circle but is nonsplit. ^[ref]