Torus action

In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus "T" is called a "T"-variety. In differential geometry, one considers an action of a real or complex torus on a manifold (or an orbifold).
A normal algebraic variety with a torus acting on it in such a way that there is a dense orbit is called a toric variety (for example, orbit closures that are normal are toric varieties).
Linear action of a torus.
A linear action of a torus can be simultaneously diagonalized, after extending the base field if necessary: if a torus "T" is acting on a finite-dimensional vector space "V", then there is a direct sum decomposition:
where
The decomposition exists because the linear action determines (and is determined by) a linear representation formula_5 and then formula_6 consists of commuting diagonalizable linear transformations, upon extending the base field.
If "V" does not have finite dimension, the existence of such a decomposition is tricky but one easy case when decomposition is possible is when "V" is a union of finite-dimensional representations (formula_7 is called rational; see below for an example). Alternatively, one uses functional analysis; for example, uses a Hilbert-space direct sum.
Example: Let formula_8 be a polynomial ring over an infinite field "k". Let formula_9 act on it as algebra automorphisms by: for formula_10
where
Then each formula_14 is a "T"-weight vector and so a monomial formula_15 is a "T"-weight vector of weight formula_16. Hence,
Note if formula_18 for all "i", then this is the usual decomposition of the polynomial ring into homogeneous components.
Białynicki-Birula decomposition.
The Białynicki-Birula decomposition says that a smooth algebraic "T"-variety admits a "T"-stable cellular decomposition.
It is often described as algebraic Morse theory.