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 ℕ be the set of non-negative integers. An ℕ-graded vector space, often called simply a graded vector space without the prefix ℕ, is a vector space V which decomposes into a direct sum of the form

$V = \bigoplus_{n \in \mathbb{N}} V_n$

where each $V_n$ is a vector space. For a given n the elements of $V_n$ 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 that can be written as a direct sum of subspaces indexed by elements i of set I:

$V = \bigoplus_{i \in I} V_i.$

Therefore, an $\mathbb{N}$-graded vector space, as defined above, is just an I-graded vector space where the set I is $\mathbb{N}$ (the set of natural numbers).

The case where I is the ring $\mathbb{Z}/2\mathbb{Z}$ (the elements 0 and 1) is particularly important in physics. A $(\mathbb{Z}/2\mathbb{Z})$-graded vector space is also known as a supervector space.

## Homomorphisms

For general index sets I, a linear map between two I-graded vector spaces f:VW 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:

$f(V_i)\subseteq W_i$ for all i in I.

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

$f(V_j)\subseteq W_{i+j}$ for all j in I,

where "+" denotes the monoid operation. If moreover I satisfies the cancellation property so that it can be embedded into a commutative group A which 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). In particular for i in I a linear map will be homogeneous of degree −i if

$f(V_{i+j})\subseteq W_j$ for all j in I, while
$f(V_j)=0\,$ if j − i is not in I.

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.

## Operations on graded vector spaces

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 VW with gradation

(VW)i = ViWi .

If I is a semigroup, then the tensor product of two I-graded vector spaces V and W is another I-graded vector space, $V \otimes W$ with gradation

$(V \otimes W)_i = \bigoplus_{\left\{\left(j,k\right)|j+k=i\right\}} V_j \otimes W_k.$