Graded vector space

Inmathematics,agraded vector spaceis avector spacethat has the extra structure of agradingorgradation,which is a decomposition of the vector space into adirect sumofvector 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.

Integer gradation

Let $\mathbb {N}$ be the set of non-negativeintegers.An ${\textstyle \mathbb {N} }$ -graded vector space,often called simply agraded vector spacewithout the prefix $\mathbb {N}$ ,is a vector space $V$ together with a decomposition into a direct sum of the form

V=\bigoplus _{n\in \mathbb {N} }V_{n}

where each $V_{n}$ is a vector space. For a givennthe elements of $V_{n}$ are then calledhomogeneouselements of degreen.

Graded vector spaces are common. For example the set of allpolynomialsin one or several variables forms a graded vector space, where the homogeneous elements of degreenare exactly the linear combinations ofmonomialsofdegreen.

General gradation

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 spaceVis a vector space together with a decomposition into a direct sum of subspaces indexed by elementsiof the setI:

V=\bigoplus _{i\in I}V_{i}.

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

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

Homomorphisms

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

f(V_{i})\subseteq W_{i}

for alliinI.

For a fixedfieldand a fixed index set, the graded vector spaces form acategorywhosemorphismsare the graded linear maps.

WhenIis acommutative monoid(such as the natural numbers), then one may more generally define linear maps that arehomogeneousof any degreeiinIby the property

f(V_{j})\subseteq W_{i+j}

for alljinI,

where "+" denotes the monoid operation. If moreoverIsatisfies thecancellation propertyso that it can beembeddedinto anabelian groupAthat it generates (for instance the integers ifIis the natural numbers), then one may also define linear maps that are homogeneous of degreeiinAby the same property (but now "+" denotes the group operation inA). Specifically, foriinIa linear map will be homogeneous of degree −iif

f(V_{i+j})\subseteq W_{j}

for alljinI,while

f(V_{j})=0\,

ifj−iis not inI.

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

Operations on graded vector spaces

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

Given twoI-graded vector spacesVandW,theirdirect sumhas underlying vector spaceV⊕ Wwith gradation

(V⊕ W)_i=V_i⊕ W_i.

IfIis asemigroup,then thetensor productof twoI-graded vector spacesVandWis anotherI-graded vector space, $V\otimes W$ ,with gradation

(V\otimes W)_{i}=\bigoplus _{\left\{\left(j,k\right)\,:\;j+k=i\right\}}V_{j}\otimes W_{k}.

Hilbert–Poincaré series

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

\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.

References

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