Graded vector space
ℕ-graded vector spaces
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
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.
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 that can be written as a direct sum of subspaces indexed by elements i of 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).
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 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.
- 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
- for all j in I, while
- 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 V ⊕ W with gradation
- (V ⊕ W)i = Vi ⊕ Wi .
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