Diagonalizable Groups

A diagonalisable group is an affine algebraic group whose coordinate ring is generated by its characters as a \(k\)-module. Equivalently: it becomes (after base change) a closed subgroup of a torus. Diagonalisable groups generalise tori to include non-connected multiplicative-type groups like \(\mu_n\) and twisted forms thereof.

Definition

Let \(G\) be an affine algebraic group over a field \(k\) (or over a base scheme). \(G\) is diagonalisable if the natural map \[ k[X^*(G)] \;\longrightarrow\; \mathcal{O}(G) \] is an isomorphism, where \(X^*(G) = \operatorname{Hom}_{k\text{-grp}}(G, \mathbb{G}_m)\) is the character group and \(k[X^*(G)]\) is its group algebra.

Equivalently: \(G\) becomes isomorphic to a closed subgroup of a torus \(\mathbb{G}_m^n\) after base change to \(\bar k\), and the embedding descends to \(k\).

Equivalent description via group of multiplicative type

A group of multiplicative type over \(k\) is the Cartier dual of a finitely generated abelian group (regarded as a constant group scheme). Specifically: for a finitely generated abelian group \(M\), the diagonalisable group \(D(M)\) has: \[ \mathcal{O}(D(M)) \;=\; k[M], \qquad D(M)(R) = \operatorname{Hom}(M, R^\times). \]

This gives an anti-equivalence of categories: \[ \{\text{diagonalisable } k\text{-groups}\} \;\longleftrightarrow\; \{\text{finitely generated abelian groups}\}. \]

Examples

\(M\)	\(D(M)\)
\(\mathbb{Z}\)	\(\mathbb{G}_m\)
\(\mathbb{Z}^n\)	\(\mathbb{G}_m^n\)
\(\mathbb{Z}/n\)	\(\mu_n\)
\(\mathbb{Z}^n \times (\mathbb{Z}/m)^k\)	\(\mathbb{G}_m^n \times \mu_m^k\)

For \(k\) of characteristic \(p\) and \(n = p\): \(D(\mathbb{Z}/p) = \mu_p\) is non-smooth (\(\mu_p\) has nilpotent functions on it). Outside characteristic \(p\), all diagonalisable groups are smooth.

Properties

Let \(G = D(M)\) be a diagonalisable group over a field \(k\):

\(G\) is affine, of multiplicative type;
\(G\) is smooth iff \(M\) has no \(p\)-torsion, where \(p = \mathrm{char}(k)\) (no condition in characteristic zero);
\(G^\circ\) (identity component) is \(D(M/M[p'])\), where \(M[p'] \subseteq M_{\mathrm{tors}}\) is the torsion of order prime to \(p\); in characteristic zero \(M[p'] = M_{\mathrm{tors}}\) and \(G^\circ\) is the maximal torus \(D(M/M_{\mathrm{tors}})\), but in characteristic \(p > 0\) the component \(G^\circ\) also contains the infinitesimal part \(D\) of the \(p\)-primary torsion (e.g. \(\mu_p\) is connected);
\(\pi_0(G) = D(M[p'])\), a finite etale group of multiplicative type;
the maximal subtorus of \(G\) is \(D(M/M_{\mathrm{tors}})\).

Twisted forms and groups of multiplicative type

A group of multiplicative type is a diagonalisable group's \(\bar k\)-form: an affine algebraic group \(G\) over \(k\) such that \(G_{\bar k}\) is diagonalisable.

Forms are classified by Galois cohomology of the automorphism group of \(M\): \[ \{\text{forms of } D(M) \text{ over } k\} \;\cong\; H^1(k, \operatorname{Aut}(M)). \]

For \(M = \mathbb{Z}^n\): forms of \(\mathbb{G}_m^n\) are exactly tori of dimension \(n\) over \(k\) (classified by \(\operatorname{GL}_n(\mathbb{Z})\)-cocycles).

Splitting

A diagonalisable group is split (over \(k\)) if \(M\) is a trivial Galois module, i.e.\ \(G\) is literally a closed subgroup of \(\mathbb{G}_m^n\) over \(k\) (not just over \(\bar k\)).

For every diagonalisable group \(G\), there exists a finite separable extension \(K/k\) such that \(G_K\) is split — the splitting field.

Functoriality

The functor \(M \mapsto D(M)\):

exact: short exact sequences in \(M\) correspond to short exact sequences in \(D(M)\);
compatible with base change: \(D(M)_S = D(M)_{\text{Spec } S}\).

This makes diagonalisable groups especially tractable: classification, classification of subgroups, classification of quotients, etc.\ all reduce to abelian-group questions.

Role in reductive structure

Maximal tori are diagonalisable groups; their character lattices and Galois actions give the root datum of a reductive group.
Centres of reductive groups are groups of multiplicative type (in general non-connected, e.g.\ \(\mu_n \subseteq \mathrm{SL}_n\)).
Diagonalizable subgroups of \(G\) are completely classified for \(G\) reductive (Borel-Steinberg).

Mathlib

Mathlib.AlgebraicGeometry.GroupScheme.Diagonalizable is in development (depends on the project's group-scheme infrastructure).
For \(\mu_n\): the project's blueprint already formalises \(\mu_n = \mathrm{Spec}\, R[\mathbb{Z}/n]\) as a Hopf-algebra.

References

Borel, Linear Algebraic Groups (1991), §8.
Springer, Linear Algebraic Groups (1998), §3.2.
SGA 3, exp.\ VIII–IX (Demazure–Grothendieck, 1970).