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).
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
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 (
Alternatively, one uses functional analysis; for example, uses a Hilbert-space direct sum.
be a polynomial ring over an infinite field k. Let
act on it as algebra automorphisms by: for
is a T-weight vector and so a monomial
for all i, then this is the usual decomposition of the polynomial ring into homogeneous components.
The Białynicki-Birula decomposition says that a smooth projective algebraic T-variety admits a T-stable cellular decomposition.
It is often described as algebraic Morse theory.
This geometry-related article is a stub.