Algebraic torus

In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by ${\displaystyle \mathbf {G} _{\mathbf {m} }}$, ${\displaystyle \mathbb {G} _{m}}$, or ${\displaystyle \mathbb {T} }$, is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups ${\displaystyle \mathbf {G} _{\mathbf {m} }}$. These groups were named by analogy with the theory of tori in Lie group theory (see Cartan subgroup). For example, over the complex numbers ${\displaystyle \mathbb {C} }$ the algebraic torus ${\displaystyle \mathbf {G} _{\mathbf {m} }}$ is isomorphic to the group scheme ${\displaystyle \mathbb {C} ^{*}={\text{Spec}}(\mathbb {C} [t,t^{-1}])}$, which is the scheme theoretic analogue of the Lie group ${\displaystyle U(1)\subset \mathbb {C} }$. In fact, any ${\displaystyle \mathbf {G} _{\mathbf {m} }}$-action on a complex vector space can be pulled back to a ${\displaystyle U(1)}$-action from the inclusion ${\displaystyle U(1)\subset \mathbb {C} ^{*}}$ as real manifolds.

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.