Inmathematics, agraded vector space is avector space that has the extra structure of agrading orgradation, which is a decomposition of the vector space into adirect sum ofvector subspaces, generally indexed by theintegers.

For "pure" vector spaces, the concept has been introduced inhomological algebra, and it is widely used forgraded algebras, which are graded vector spaces with additional structures.

Let${\displaystyle \mathbb{N}}$ be the set of non-negativeintegers. An$\mathbb{N}$-graded vector space, often called simply agraded vector space without the prefix${\displaystyle \mathbb{N}}$, is a vector spaceV together with a decomposition into a direct sum of the form

${\displaystyle V=\underset{n\in \mathbb{N}}{⨁}{V}_n}$

where each${\displaystyle V_n}$ is a vector space. For a givenn the elements of${\displaystyle V_n}$ are then calledhomogeneous elements of degreen.

Graded vector spaces are common. For example the set of allpolynomials in one or several variables forms a graded vector space, where the homogeneous elements of degreen are exactly the linear combinations ofmonomials ofdegree 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 setI. AnI-graded vector spaceV is a vector space together with a decomposition into a direct sum of subspaces indexed by elementsi of the setI:

${\displaystyle V=\underset{i\in I}{⨁}{V}_i.}$

Therefore, an${\displaystyle \mathbb{N}}$-graded vector space, as defined above, is just anI-graded vector space where the setI is${\displaystyle \mathbb{N}}$ (the set ofnatural numbers).

The case whereI is thering${\displaystyle \mathbb{Z}/2\mathbb{Z}}$ (the elements 0 and 1) is particularly important inphysics. A${\displaystyle (\mathbb{Z}/2\mathbb{Z})}$-graded vector space is also known as asupervector space.

For general index setsI, alinear map between twoI-graded vector spacesf :V →W is called agraded linear map if it preserves the grading of homogeneous elements. A graded linear map is also called ahomomorphism (ormorphism) of graded vector spaces, orhomogeneous linear map:

${\displaystyle f(V_i)\subseteq W_i}$ for alli inI.

For a fixedfield and a fixedindex set, the graded vector spaces form acategory whosemorphisms are the graded linear maps.

WhenI is acommutativemonoid (such as the natural numbers), then one may more generally define linear maps that arehomogeneous of any degreei inI by the property

${\displaystyle f(V_j)\subseteq W_{i+j}}$ for allj inI,

where "+" denotes the monoid operation. If moreoverI satisfies thecancellation property so that it can beembedded into anabelian groupA that it generates (for instance the integers ifI is the natural numbers), then one may also define linear maps that are homogeneous of degreei inA by the same property (but now "+" denotes thegroup operation inA). Specifically, fori inI a linear map will be homogeneous of degree −i if

${\displaystyle f(V_{i+j})\subseteq W_j}$ for allj inI, while

${\displaystyle f(V_j)=0}$ ifj −i is not inI.

Just as the set of linear maps from a vector space to itself forms anassociative algebra (thealgebra of endomorphisms of the vector space), the sets of homogeneous linear maps from a space to itself – either restricting degrees toI or allowing any degrees in the groupA – form associativegraded algebras over those index sets.

Some operations on vector spaces can be defined for graded vector spaces as well.

Given twoI-graded vector spacesV andW, theirdirect sum has underlying vector spaceV ⊕ W with gradation

(V ⊕ W)_i =V_i ⊕ W_i .

IfI is asemigroup, then thetensor product of twoI-graded vector spacesV andW is anotherI-graded vector space,${\displaystyle V\otimes W}$, with gradation

${\displaystyle (V\otimes W{)}_i=\underset{\left\{\left(j,k\right):j+k=i\right\}}{⨁}{V}_j\otimes W_k.}$

Given a${\displaystyle \mathbb{N}}$-graded vector space that is finite-dimensional for every${\displaystyle n\in \mathbb{N},}$ itsHilbert–Poincaré series is theformal power series

${\displaystyle \sum _{n\in \mathbb{N}}{\dim }_K(V_n)t^n.}$

From the formulas above, the Hilbert–Poincaré series of a direct sum and of a tensor product of graded vector spaces (finite dimensional in each degree) are respectively the sum and the product of the corresponding Hilbert–Poincaré series.

• Graded (mathematics)
• Graded algebra
• Comodule
• Graded module
• Littlewood–Richardson rule

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

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• Vector spaces

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