In mathematics, a graded vector space is a vector space that has the extra structure of a grading or a gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.
Let be the set of non-negative integers. An -graded vector space, often called simply a graded vector space without the prefix , is a vector space V together with a decomposition into a direct sum of the form
where each is a vector space. For a given n the elements of are then called homogeneous elements of degree n.
Graded vector spaces are common. For example the set of all polynomials in one or several variables forms a graded vector space, where the homogeneous elements of degree n are exactly the linear combinations of monomials of degree n.
The subspaces of a graded vector space need not be indexed by the set of natural numbers, and may be indexed by the elements of any set I. An I-graded vector space V is a vector space together with a decomposition into a direct sum of subspaces indexed by elements i of the set I:
Therefore, an -graded vector space, as defined above, is just an I-graded vector space where the set I is (the set of natural numbers).
The case where I is the ring (the elements 0 and 1) is particularly important in physics. A -graded vector space is also known as a supervector space.
For general index sets I, a linear map between two I-graded vector spaces f : V → W is called a graded linear map if it preserves the grading of homogeneous elements. A graded linear map is also called a homomorphism (or morphism) of graded vector spaces, or homogeneous linear map:
For a fixed field and a fixed index set, the graded vector spaces form a category whose morphisms are the graded linear maps.
When I is a commutative monoid (such as the natural numbers), then one may more generally define linear maps that are homogeneous of any degree i in I by the property
where "+" denotes the monoid operation. If moreover I satisfies the cancellation property so that it can be embedded into an abelian group A that it generates (for instance the integers if I is the natural numbers), then one may also define linear maps that are homogeneous of degree i in A by the same property (but now "+" denotes the group operation in A). Specifically, for i in I a linear map will be homogeneous of degree −i if
Just as the set of linear maps from a vector space to itself forms an associative algebra (the algebra of endomorphisms of the vector space), the sets of homogeneous linear maps from a space to itself – either restricting degrees to I or allowing any degrees in the group A – form associative graded algebras over those index sets.
Some operations on vector spaces can be defined for graded vector spaces as well.
Given two I-graded vector spaces V and W, their direct sum has underlying vector space V ⊕ W with gradation
If I is a semigroup, then the tensor product of two I-graded vector spaces V and W is another I-graded vector space, , with gradation
Given a -graded vector space that is finite-dimensional for every its Hilbert–Poincaré series is the formal power series
From the formulas above, the Hilbert–Poincaré series of a direct sum and of a tensor product of graded vector spaces (finite dimensional in each degree) are respectively the sum and the product of the corresponding Hilbert–Poincaré series.