Questions tagged [reductive-groups]

Reductive groups form a large class of linear algebraic groups interpolating between semisimple algebraic groups, like $\mathrm{SL}_n$, and multiplicative algebraic groups, like $\mathbb{G}_m$, and including the most important algebraic group of all, $\mathrm{GL}_n$.

There are many way of defining reductive groups. One could use:

• the unipotent radical: An algebraic group $G$ is reductive if the largest normal unipotent subgroup $R_u(G)$ is trivial
• representations: An algebraic group $G$ is reductive if its category of representations is semisimple.
• group quotients: If $G$ is a linear algebraic group, then it has an embedding $G \hookrightarrow \mathrm{GL}_n$; if $\mathrm{GL}_n/G$ is an affine scheme, then $G$ is reductive.

Reductive groups can be studied analogously to semisimple Lie groups. If $T \subset G$ is a maximal split torus (meanining $T$ is the largest multiplicative subgroup of $G$ isomorphic to a cartesian product of $\mathbb{G}_m$), then the Lie algebra $\mathfrak{g}$ of $G$ decomposes into root spaces:

$$ \mathfrak{g} = \mathfrak{t} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}, $$

where $\mathop{Lie}(T) = \mathfrak{t}$ acts on $\mathfrak{g}_{\alpha}$ via the root $\alpha$. Also, the irreducible representations of $G$ are highest-weight representations, classified by dominant weights, or morphisms $T \to \mathbb{G}_m$.

Furthermore, the classification of reductive groups is closely related to the classification of semisimple Lie algebras, with some slight complication. If $G$ is a simple reductive group defined over an algebraically closed field, then its root system $\Phi$ is a member of one of $4$ infinite families $A_n$, $B_n$, $C_n$, or $D_n$, or else is one of $5$ exceptional types $E_6$, $E_7$, $E_8$, $F_4$, or $G_2$—the exact same $7$ families for root systems of Lie algebras. The full classification of reductive groups also takes into account the lattices of coweights $X^{\vee} = \mathrm{Hom}(\mathbb{G}_m, T)$ and weights $X = \mathrm{Hom}(T, \mathbb{G}_m)$, forming a root datum $(X, \Phi, X^{\vee})$. For each possible root datum there is exactly one connected reductive group over a given algebraically closed field, up to isomorphism.