Graded vector space

<h2 id="integer-gradation">Integer gradation</h2>
Let 
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbb {N} }
 
 be the set of non-negative <a href="/facts/Integer/PUwwrawx">integers</a>. An 
 
 
 
 
 N
 
 
 
 {\textstyle \mathbb {N} }
 
-graded vector space, often called simply a graded vector space without the prefix 
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbb {N} }
 
, is a vector space V together with a decomposition into a direct sum of the form

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

where each 
 
 
 
 
 V
 
 n
 
 
 
 
 {\displaystyle V_{n}}
 
 is a vector space. For a given n the elements of 
 
 
 
 
 V
 
 n
 
 
 
 
 {\displaystyle V_{n}}
 
 are then called homogeneous elements of degree n.
Graded vector spaces are common. For example the set of all <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> in one or several variables forms a graded vector space, where the homogeneous elements of degree n are exactly the linear combinations of <a href="/facts/Monomial/6VAPCYXJ">monomials</a> of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> n.

<h2 id="general-gradation">General gradation</h2>
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:

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

Therefore, an 
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbb {N} }
 
-graded vector space, as defined above, is just an I-graded vector space where the set I is 
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbb {N} }
 
 (the set of <a href="/facts/Natural_number/u65og8JE">natural numbers</a>).
The case where I is the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> 
 
 
 
 
 Z
 
 
 /
 
 2
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} /2\mathbb {Z} }
 
 (the elements 0 and 1) is particularly important in <a href="/facts/Physics/a7aH2y08">physics</a>. A 
 
 
 
 (
 
 Z
 
 
 /
 
 2
 
 Z
 
 )
 
 
 {\displaystyle (\mathbb {Z} /2\mathbb {Z} )}
 
-graded vector space is also known as a <a href="/facts/Supervector_space/JW8jElFE">supervector space</a>.

<h2 id="homomorphisms">Homomorphisms</h2>

"Homogeneous linear map" redirects here. For the more general concept, see <a href="/facts/Graded_module_homomorphism/AIUjkSaP">Graded module homomorphism</a>.
For general index sets I, a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> 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:

f
 (
 
 V
 
 i
 
 
 )
 ⊆
 
 W
 
 i
 
 
 
 
 {\displaystyle f(V_{i})\subseteq W_{i}}
 
 for all i in I.
For a fixed <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> and a fixed index set, the graded vector spaces form a <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> whose <a href="/facts/Morphism/Kdpuyz3S">morphisms</a> are the graded linear maps.
When I is a <a href="/facts/Commutative_monoid/6AHve7p8">commutative</a> <a href="/facts/Monoid/6AHve7p8">monoid</a> (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

f
 (
 
 V
 
 j
 
 
 )
 ⊆
 
 W
 
 i
 +
 j
 
 
 
 
 {\displaystyle f(V_{j})\subseteq W_{i+j}}
 
 for all j in I,
where "+" denotes the monoid operation. If moreover I satisfies the <a href="/facts/Cancellation_property/0Cd1rKPk">cancellation property</a> so that it can be <a href="/facts/Embedding/HKeVKc42">embedded</a> into an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> 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

f
 (
 
 V
 
 i
 +
 j
 
 
 )
 ⊆
 
 W
 
 j
 
 
 
 
 {\displaystyle f(V_{i+j})\subseteq W_{j}}
 
 for all j in I, while

f
 (
 
 V
 
 j
 
 
 )
 =
 0
 
 
 
 {\displaystyle f(V_{j})=0\,}
 
 if j − i is not in I.
Just as the set of linear maps from a vector space to itself forms an <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> (the <a href="/facts/Endomorphism_algebra/K1UgpnJK">algebra of endomorphisms</a> 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 <a href="/facts/Graded_algebra/AIUjkSaP">graded algebras</a> over those index sets.

<h2 id="operations-on-graded-vector-spaces">Operations on graded vector spaces</h2>
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 = Vi ⊕ Wi .
If I is a <a href="/facts/Semigroup/rgSdk2kt">semigroup</a>, then the tensor product of two I-graded vector spaces V and W is another I-graded vector space, 
 
 
 
 V
 ⊗
 W
 
 
 {\displaystyle V\otimes W}
 
, with gradation

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

<h2 id="hilbertpoincar-series">Hilbert–Poincaré series</h2>
Given a 
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbb {N} }
 
-graded vector space that is finite-dimensional for every 
 
 
 
 n
 ∈
 
 N
 
 ,
 
 
 {\displaystyle n\in \mathbb {N} ,}
 
 its <a href="/facts/Hilbert%E2%80%93Poincar%C3%A9_series/cVJBU8yR">Hilbert–Poincaré series</a> is the <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a>

∑
          
            n
            ∈
            
              N
            
          
        
        
          dim
          
            K
          
        
        ⁡
        (
        
          V
          
            n
          
        
        )
        
        
          t
          
            n
          
        
        .
      
    
    {\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.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Graded_(mathematics)/K0rghWx6">Graded (mathematics)</a></li>
<li><a href="/facts/Graded_algebra/AIUjkSaP">Graded algebra</a></li>
<li><a href="/facts/Comodule/AXrEw6gX">Comodule</a></li>
<li><a href="/facts/Graded_module/AIUjkSaP">Graded module</a></li>
<li><a href="/facts/Littlewood%E2%80%93Richardson_rule/8NSx5K36">Littlewood–Richardson rule</a></li></ul>

<ul><li><a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki, N.</a> (1974) Algebra I (Chapters 1-3), <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-64243-5, Chapter 2, Section 11; Chapter 3.</li></ul>

Graded vector space open-in-new

Graded vector space