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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]

For an algebraic torus $T$ over $k$ and a chosen separable closure $k_s$, its geometric character lattice of an algebraic torus isIt is a free abelian group of rank $\dim T$, with its natural action of $\mathrm{Gal}(k_s/k)$. Characters of an algebraic torus defined over $k$ form the fixed subgroup of this action; they need not constitute the full geometric toric lattice.

For a split algebraic torus $T={\mathbb{G}}_m^n$, each algebraic torus character is a Laurent monomial $t_1^{m_1}\cdots t_n^{m_n}$, so $X^{\ast }(T)\cong {\mathbb{Z}}^n$. The real norm-one algebraic torus has geometric algebraic torus character toric lattice $\mathbb{Z}$ with complex conjugation acting by minus one, so its subgroup of real-defined characters of an algebraic torus is zero.^[ref]

The geometric cocharacter lattice of an algebraic torus $T$ over $k$ isusing a separable closure $k_s$. Composition with an algebraic torus character gives a homomorphism ${\mathbb{G}}_m\to {\mathbb{G}}_m$, necessarily $z\mapsto z^a$ for an integer $a$. This defines the perfect pairingFor a split algebraic torus, the cocharacter with exponent vector $(n_1,\dots ,n_r)$ is $z\mapsto (z^{n_1},\dots ,z^{n_r})$ and the pairing is ${\sum }_i{m}_i{n}_i$. The two geometric toric lattices carry compatible Galois group actions; $k$-defined cocharacters are the fixed elements. This is the duality used for the toric lattice $N$ and its dual $M$.^[ref]

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]

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]