Algebraic torus

In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of tori in Lie group theory (see Cartan subgroup). In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of tori in Lie group theory (see Cartan subgroup). Tori are of fundamental importance in the theory of algebraic groups and Lie groups and in the study of the geometric objects associated to them such as symmetric spaces and buildings. In most places we suppose that the base field is perfect (for example finite or characteristic zero). In general one has to use separable closures instead of algebraic closures. If F {displaystyle F} is a field then the multiplicative group over F {displaystyle F} is the algebraic group G m {displaystyle mathbf {G} _{mathbf {m} }} such that for any field extension E / F {displaystyle E/F} the E {displaystyle E} -points are isomorphic to the group E × {displaystyle E^{ imes }} . To define it properly as an algebraic group one can take the affine variety defined by the equation x y = 1 {displaystyle xy=1} in the affine plane over F {displaystyle F} with coordinates x , y {displaystyle x,y} . The multiplication is then given by restricting the regular rational map F 2 × F 2 → F 2 {displaystyle F^{2} imes F^{2} o F^{2}} defined by ( ( x , y ) , ( x ′ , y ′ ) ) ↦ ( x x ′ , y y ′ ) {displaystyle ((x,y),(x',y'))mapsto (xx',yy')} and the inverse is the restriction of the regular rational map ( x , y ) ↦ ( y , x ) {displaystyle (x,y)mapsto (y,x)} . Let F {displaystyle F} be a field with algebraic closure F ¯ {displaystyle {overline {F}}} . Then a F {displaystyle F} -torus is an algebraic group defined over F {displaystyle F} which is isomorphic over F ¯ {displaystyle {overline {F}}} to a finite product of copies of the multiplicative group. In other words, if T {displaystyle mathbf {T} } is an F {displaystyle F} -group it is a torus if and only if T ( F ¯ ) ≅ ( F ¯ × ) r {displaystyle mathbf {T} ({overline {F}})cong ({overline {F}}^{ imes })^{r}} for some r ≥ 1 {displaystyle rgeq 1} . The basic terminology associated to tori is as follows. An isogeny between algebraic groups is a surjective morphism with finite kernel; two tori are said to be isogenous if there exists an isogeny from the first to the second. Isogenies between tori are particularly well-behaved: for any isogeny ϕ : T → T ′ {displaystyle phi :mathbf {T} o mathbf {T} '} there exists a 'dual' isogeny ψ : T ′ → T {displaystyle psi :mathbf {T} ' o mathbf {T} } such that ψ ∘ ϕ {displaystyle psi circ phi } is a power map. In particular being isogenous is an equivalence relation between tori.