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 nthe 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.
General ''I''-graded vector spaces
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 set the I: Therefore, an -graded vector space, as defined above, is just an I-graded vector space where the set I is . The case where I is the ring is particularly important in physics. A -graded vector space is also known as a supervector space.
Homomorphisms
For general indexsetsI, a linear map between two I-graded vector spaces is called a graded linear map if it preserves the grading of homogeneous elements. A graded linear map is also called a homomorphism of graded vector spaces, or homogeneous linear map: For a fixed field and a fixedindex set, the graded vector spaces form a category whose morphisms are the graded linear maps. When I is a commutative monoid, then one may more generally define linear maps that are homogeneous of any degree i in I by the property where "+" denotes the monoidoperation. If moreover I satisfies the cancellation property so that it can be embedded into a commutative groupA that it generates, then one may also define linear maps that are homogeneous of degree i in A by the same property. 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 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 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
