# Graded vector space

In mathematics, a **graded vector space** is a type of vector space that includes the extra structure of **gradation**, which is a decomposition of the vector space into a direct sum of vector subspaces.

## N-graded vector spaces

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* 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 variable form 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).

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.

## Linear maps

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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:

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 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

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}=*V*⊕_{i}*W*._{i}

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

## See also

## References

- Bourbaki, N. (1974)
*Algebra I*(Chapters 1-3), ISBN 978-3-540-64243-5, Chapter 2, Section 11; Chapter 3.