In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by , , or , 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 . These groups were named by analogy with the theory of tori in Lie group theory (see Cartan subgroup). For example, over the complex numbers the algebraic torus is isomorphic to the group scheme , which is the scheme theoretic analogue of the Lie group . In fact, any -action on a complex vector space can be pulled back to a -action from the inclusion 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.
In mathematics, an algebraictorus, where a one dimensional torus is typically denoted by G m {\displaystyle \mathbf {G} _{\mathbf {m} }} , G m {\displaystyle...
In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraictorus as an open dense subset, such that the...
natural number n. For a linear algebraic group G, a maximal torus in G means a torus in G that is not contained in any bigger torus. For example, the group of...
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as...
conjecture Algebraic geometry and analytic geometry Mirror symmetry Linear algebraic group Additive group Multiplicative group Algebraictorus Reductive...
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants...
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies...
element of F and the binary operation • is the field multiplication, the algebraictorus GL(1).[clarification needed]. The multiplicative group of integers...
mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus...
In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraictorus on the variety. A variety equipped with an action...
the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup. For example...
Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group G is a compact, connected...
least twice in order for them all to cancel. For X an algebraic variety in the algebraictorus ( K × ) n {\displaystyle (K^{\times })^{n}} , the tropical...
that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined geometrically, as an...
not algebraic. On the other hand, every projective complex manifold is necessarily algebraic, see Chow's theorem. As an example, consider the torus T := C/(Z + τ...
C*-algebras which generalize the algebra of continuous functions on the 2-torus. Many topological and geometric properties of the classical 2-torus have...
can be phrased in the language of algebraic groups, in terms of the existence of a group action of an algebraictorus (equivalently by a multiplicative...
n {\displaystyle {\mathfrak {gl}}_{n}} , but is not necessarily an algebraictorus). If the matrix M {\displaystyle M} is diagonalisable, then it is regular...
theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group G {\displaystyle G} over a (not necessarily algebraically closed) field...
of an abelian variety by a linear group. If this linear group is an algebraictorus, so that A k 0 {\displaystyle A_{k}^{0}} is a semiabelian variety,...
geometry, a pinched torus (or croissant surface) is a kind of two-dimensional surface. It gets its name from its resemblance to a torus that has been pinched...
In algebraic geometry, a tropical compactification is a compactification (projective completion) of a subvariety of an algebraictorus, introduced by...
In mathematics, a genus g surface (also known as a g-torus or g-holed torus) is a surface formed by the connected sum of g distinct tori: the interior...
diagonalizable group is called an algebraictorus (which is not necessarily compact, in contrast to a complex torus). A k-torus is a torus defined over k. The centralizer...
reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive...
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![{\displaystyle \mathbb {C} ^{*}={\text{Spec}}(\mathbb {C} [t,t^{-1}])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03ed370c7c49bbe49f8570f0ccd785b881291ee1)
