Editing Graded vector space

{{Short description|Algebraic structure decomposed into a direct sum}}
{{Use American English|date = February 2019}}
In [[mathematics]], a '''graded vector space''' is a [[vector space]] that has the extra structure of a ''grading'' or ''gradation'', which is a decomposition of the vector space into a [[direct sum of vector spaces|direct sum]] of [[linear subspace|vector subspaces]], generally indexed by the [[integer]]s.

For "pure" vector spaces, the concept has been introduced in [[homological algebra]], and it is widely used for [[graded algebra]]s, which are graded vector spaces with additional structures.

== Integer gradation ==
Let <math>\mathbb{N}</math> be the set of non-negative [[integer]]s. An <math display="inline">\mathbb{N}</math>'''-graded vector space''', often called simply a '''graded vector space''' without the prefix <math>\mathbb{N}</math>, is a vector space {{math|''V''}} together with a decomposition into a direct sum of the form

: <math>V = \bigoplus_{n \in \mathbb{N}} V_n</math>
where each <math>V_n</math> is a vector space.  For a given ''n'' the elements of <math>V_n</math> are then called '''homogeneous''' elements of degree ''n''.

Graded vector spaces are common.  For example the set of all [[polynomial]]s in one or several variables forms a graded vector space, where the homogeneous elements of degree ''n'' are exactly the linear combinations of [[monomial]]s of [[degree of a polynomial|degree]]&nbsp;''n''.

==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 set ''I''.  An ''I''-graded vector space ''V'' is a vector space together with a decomposition into a direct sum of subspaces indexed by elements ''i'' of the set ''I'':
: <math>V = \bigoplus_{i \in I} V_i.</math>

Therefore, an <math>\mathbb{N}</math>-graded vector space, as defined above, is just an ''I''-graded vector space where the set ''I'' is <math>\mathbb{N}</math> (the set of [[natural number]]s).

The case where ''I'' is the [[ring (mathematics)|ring]] <math>\mathbb{Z}/2\mathbb{Z}</math> (the elements 0 and 1) is particularly important in [[physics]]. A <math>(\mathbb{Z}/2\mathbb{Z})</math>-graded vector space is also known as a [[supervector space]].

==Homomorphisms==
{{Anchor|Graded linear map|Linear maps}}
{{redirect|Homogeneous linear map|the more general concept|Graded module homomorphism}}
For general index sets ''I'', a [[linear map]] between two ''I''-graded vector spaces {{nowrap|''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''':

:<math>f(V_i)\subseteq W_i</math>   for all ''i'' in ''I''.

For a fixed [[field (mathematics)|field]] and a fixed index set, the graded vector spaces form a [[category (mathematics)|category]] whose [[morphism]]s are the graded linear maps.

When ''I'' is a [[commutative monoid|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

:<math>f(V_j)\subseteq W_{i+j}</math>   for all ''j'' in ''I'',

where "+" denotes the monoid operation. If moreover ''I'' satisfies the [[cancellation property]] so that it can be [[embedding|embedded]] into an [[abelian group]] ''A'' that 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''). Specifically, for ''i'' in ''I'' a linear map will be homogeneous of degree −''i'' if

:<math>f(V_{i+j})\subseteq W_j</math>   for all ''j'' in ''I'', while
:<math>f(V_j)=0\,</math>   if {{nowrap|''j'' − ''i''}} is not in ''I''.

Just as the set of linear maps from a vector space to itself forms an [[associative algebra]] (the [[endomorphism algebra|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 algebra]]s over those index sets.

==<span id="directsum"></span><span id="tensorproduct"></span>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''&nbsp;⊕&thinsp;''W'' with gradation
:(''V''&nbsp;⊕&thinsp;''W'')<sub>''i''</sub> = ''V<sub>i</sub>''&nbsp;⊕&thinsp;''W<sub>i</sub>''&nbsp;.

If ''I'' is a [[semigroup]], then the '''tensor product''' of two ''I''-graded vector spaces ''V'' and ''W'' is another ''I''-graded vector space, <math>V \otimes W</math>, with gradation
: <math>(V \otimes W)_i = \bigoplus_{\left\{\left(j,k\right) \,:\; j+k=i\right\}} V_j \otimes W_k.</math>

==Hilbert–Poincaré series==

Given a <math>\N</math>-graded vector space that is finite-dimensional for every <math>n\in \N,</math> its [[Hilbert–Poincaré series]] is the [[formal power series]]
:<math>\sum_{n\in\N}\dim_K(V_n)\, t^n.</math>

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.

==See also==
* [[Graded (mathematics)]]
* [[Graded algebra]]
* [[Comodule]]
* [[Graded module]]
* [[Littlewood–Richardson rule]]

==References==
* [[Nicolas Bourbaki|Bourbaki, N.]] (1974) ''Algebra I'' (Chapters 1-3), {{ISBN|978-3-540-64243-5}}, Chapter 2, Section 11; Chapter 3.

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[[Category:Categories in category theory]]
[[Category:Vector spaces]]