Affine algebraic group

An affine algebraic group is an affine variety $G$ whose multiplication and inversion are regular maps. Equivalently, its coordinate ring $k[G]$ is a commutative Hopf algebra, with comultiplication induced by multiplication in $G$.

Every affine algebraic group has a faithful finite-dimensional rational group representation. Choose algebra generators of $k[G]$ and place them in a finite-dimensional subspace stable under right translations. An element acting trivially on that subspace has the same coordinate values as the identity and therefore is the identity.

The group $G\times G$ acts on $k[G]$ byFor $G={\mathbb{G}}_m$, this gives $k[G]={⨁}_{r\in \mathbb{Z}}{k}_{(-r,r)}$. For $G={\mathbb{G}}_a$, the action factors through $(a,b)\mapsto b-a$ and the degree filtration of $k[t]$ has trivial one-dimensional successive quotients.

An element of an affine algebraic group is semisimple when its image in one, equivalently every, faithful finite-dimensional representation is a diagonalizable linear map.

An element of an affine algebraic group is unipotent when its image in one, equivalently every, faithful finite-dimensional representation has every eigenvalue equal to one.

Every element $g$ of an affine algebraic group has unique commuting semisimple and unipotent parts $g_s,g_u$ such that $g=g_s{g}_u$. The construction agrees with the multiplicative Jordan decomposition in every rational representation.

The derived subgroup $[G,G]$ is the closed algebraic subgroup generated by the commutators $xy{x}^{-1}{y}^{-1}$. If $G$ is connected, images of finite products of commutators are connected and their closures stabilize by dimension, which shows that $[G,G]$ is connected.

An affine algebraic group is solvable when its derived series reaches the identity. A connected solvable affine algebraic group can be conjugated into an upper triangular matrix group by the Lie-Kolchin theorem.

Every connected solvable affine algebraic subgroup of $\mathrm{GL}(V)$ preserves a complete flag in $V$, equivalently it is conjugate to a subgroup of the upper triangular matrices.

A unipotent algebraic group is an affine algebraic group all of whose elements are unipotent. The Kolchin theorem embeds it into an upper unitriangular group.

Every unipotent algebraic subgroup of ${\mathrm{GL}}_n$ is conjugate to a subgroup of the upper unitriangular group. The filtration by vanishing initial superdiagonals then proves that every unipotent algebraic group is a nilpotent group.

A diagonalizable algebraic group is isomorphic to a closed subgroup of a product of copies of ${\mathbb{G}}_m$. Every rational representation of such a group decomposes into weight spaces.

A reductive algebraic group is a smooth connected affine algebraic group whose largest connected normal unipotent algebraic group is trivial.

A Borel subgroup is a maximal closed connected solvable subgroup of an affine algebraic group.

A parabolic subgroup of a connected reductive group is a closed subgroup containing a Borel subgroup. Its quotient in the group is projective.

A Levi subgroup is a reductive complement to the unipotent radical of a parabolic subgroup.

For a connected reductive algebraic group $G$ and a Borel subgroup $B$, the quotient $G/B$ is its complete flag variety. Quotients $G/P$ by parabolic subgroups are partial flag varieties.

A root datum is a quadruple $(X^{\ast },\Phi ,X_{\ast },{\Phi }^{\vee })$ of dual lattices, roots, and coroots with the natural perfect pairing and reflection axioms. A reductive group with maximal torus $T$ has $X^{\ast }=X^{\ast }(T)$ and $X_{\ast }=X_{\ast }(T)$.

For a connected reductive algebraic group, a Borel subgroup $B$, and a maximal torus $T\subseteq B$, the Weyl group $W=N_G(T)/T$ indexes the double cosets:

For an algebraic group $H$, a flat $H$-torsor over $Y$ is a faithfully flat morphism $X\to Y$ with a right $H$-action such that $X\times H\to X{\times }_{Y}X$, $(x,h)\mapsto (x,xh)$, is an isomorphism. A Zariski torsor additionally becomes $U\times H$ on a Zariski open cover of $Y$.