# Graded vector space

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.

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## ℕ-graded vector spaces[edit]

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[edit]

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.

## Homomorphisms[edit]

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.

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

- 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[edit]

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[edit]

- Graded (mathematics)
- Graded algebra
- Hilbert–Poincaré series
- Comodule
- Graded module
- Littlewood–Richardson rule

## References[edit]

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