Alternating multilinear map

<h2 id="definition">Definition</h2>
<p>Let 
  
    
      
        R
      
    
    {\displaystyle R}
  
 be a commutative ring and 
  
    
      
        V
      
    
    {\displaystyle V}
  
, 
  
    
      
        W
      
    
    {\displaystyle W}
  
 be modules over 
  
    
      
        R
      
    
    {\displaystyle R}
  
. A multilinear map of the form 
  
    
      
        f
        :
        
          V
          
            n
          
        
        →
        W
      
    
    {\displaystyle f:V^{n}\to W}
  
 is said to be alternating if it satisfies the following equivalent conditions:
</p>
<ol><li>whenever there exists 
  
    
      
        1
        ≤
        i
        ≤
        n
        −
        1
      
    
    {\textstyle 1\leq i\leq n-1}
  
 such that 
  
    
      
        
          x
          
            i
          
        
        =
        
          x
          
            i
            +
            1
          
        
      
    
    {\displaystyle x_{i}=x_{i+1}}
  
 then 
  
    
      
        f
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        =
        0
      
    
    {\displaystyle f(x_{1},\ldots ,x_{n})=0}
  
.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li>
<li>whenever there exists 
  
    
      
        1
        ≤
        i
        ≠
        j
        ≤
        n
      
    
    {\textstyle 1\leq i\neq j\leq n}
  
 such that 
  
    
      
        
          x
          
            i
          
        
        =
        
          x
          
            j
          
        
      
    
    {\displaystyle x_{i}=x_{j}}
  
 then 
  
    
      
        f
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        =
        0
      
    
    {\displaystyle f(x_{1},\ldots ,x_{n})=0}
  
.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li></ol>
<h2 id="vector-spaces">Vector spaces</h2>
<p>Let 
  
    
      
        V
        ,
        W
      
    
    {\displaystyle V,W}
  
 be vector spaces over the same field. Then a multilinear map of the form 
  
    
      
        f
        :
        
          V
          
            n
          
        
        →
        W
      
    
    {\displaystyle f:V^{n}\to W}
  
 is alternating if it satisfies the following condition:
</p>
<ul><li>if 
  
    
      
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
      
    
    {\displaystyle x_{1},\ldots ,x_{n}}
  
 are <a href="/facts/Linearly_dependent/8JrHfIIa">linearly dependent</a> then 
  
    
      
        f
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        =
        0
      
    
    {\displaystyle f(x_{1},\ldots ,x_{n})=0}
  
.</li></ul>
<h2 id="example">Example</h2>
<p>In a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a>, the <a href="/facts/Lie_algebra/n2gAUkNq">Lie bracket</a> is an alternating bilinear map. 
The <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of a matrix is a multilinear alternating map of the rows or columns of the matrix.
</p>
<h2 id="properties">Properties</h2>
<p>If any component 
  
    
      
        
          x
          
            i
          
        
      
    
    {\displaystyle x_{i}}
  
 of an alternating multilinear map is replaced by 
  
    
      
        
          x
          
            i
          
        
        +
        c
        
          x
          
            j
          
        
      
    
    {\displaystyle x_{i}+cx_{j}}
  
 for any 
  
    
      
        j
        ≠
        i
      
    
    {\displaystyle j\neq i}
  
 and 
  
    
      
        c
      
    
    {\displaystyle c}
  
 in the base <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> 
  
    
      
        R
      
    
    {\displaystyle R}
  
, then the value of that map is not changed.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p><p>Every alternating multilinear map is antisymmetric,<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> meaning that<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>

f
        (
        …
        ,
        
          x
          
            i
          
        
        ,
        
          x
          
            i
            +
            1
          
        
        ,
        …
        )
        =
        −
        f
        (
        …
        ,
        
          x
          
            i
            +
            1
          
        
        ,
        
          x
          
            i
          
        
        ,
        …
        )
        
        
           for any 
        
        1
        ≤
        i
        ≤
        n
        −
        1
        ,
      
    
    {\displaystyle f(\dots ,x_{i},x_{i+1},\dots )=-f(\dots ,x_{i+1},x_{i},\dots )\quad {\text{ for any }}1\leq i\leq n-1,}

or equivalently,

f
        (
        
          x
          
            σ
            (
            1
            )
          
        
        ,
        …
        ,
        
          x
          
            σ
            (
            n
            )
          
        
        )
        =
        (
        sgn
        ⁡
        σ
        )
        f
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        
        
           for any 
        
        σ
        ∈
        
          
            S
          
          
            n
          
        
        ,
      
    
    {\displaystyle f(x_{\sigma (1)},\dots ,x_{\sigma (n)})=(\operatorname {sgn} \sigma )f(x_{1},\dots ,x_{n})\quad {\text{ for any }}\sigma \in \mathrm {S} _{n},}

where 
  
    
      
        
          
            S
          
          
            n
          
        
      
    
    {\displaystyle \mathrm {S} _{n}}
  
 denotes the <a href="/facts/Permutation_group/vRsNNwwx">permutation group</a> of degree 
  
    
      
        n
      
    
    {\displaystyle n}
  
 and 
  
    
      
        sgn
        ⁡
        σ
      
    
    {\displaystyle \operatorname {sgn} \sigma }
  
 is the <a href="/facts/Parity_of_a_permutation/ludhCMJz">sign</a> of 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
If 
  
    
      
        n
        !
      
    
    {\displaystyle n!}
  
 is a <a href="/facts/Unit_(ring_theory)/GJSBu8Y7">unit</a> in the base ring 
  
    
      
        R
      
    
    {\displaystyle R}
  
, then every antisymmetric 
  
    
      
        n
      
    
    {\displaystyle n}
  
-multilinear form is alternating.
</p>
<h2 id="alternatization">Alternatization</h2>
<p>Given a multilinear map of the form 
  
    
      
        f
        :
        
          V
          
            n
          
        
        →
        W
        ,
      
    
    {\displaystyle f:V^{n}\to W,}
  
 the alternating multilinear map 
  
    
      
        g
        :
        
          V
          
            n
          
        
        →
        W
      
    
    {\displaystyle g:V^{n}\to W}
  
 defined by

g
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        
          :=
        
        
          ∑
          
            σ
            ∈
            
              S
              
                n
              
            
          
        
        sgn
        ⁡
        (
        σ
        )
        f
        (
        
          x
          
            σ
            (
            1
            )
          
        
        ,
        …
        ,
        
          x
          
            σ
            (
            n
            )
          
        
        )
      
    
    {\displaystyle g(x_{1},\ldots ,x_{n})\mathrel {:=} \sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )f(x_{\sigma (1)},\ldots ,x_{\sigma (n)})}

is said to be the alternatization of 
  
    
      
        f
      
    
    {\displaystyle f}
  
.
</p><p>Properties
</p>
<ul><li>The alternatization of an 
  
    
      
        n
      
    
    {\displaystyle n}
  
-multilinear alternating map is 
  
    
      
        n
        !
      
    
    {\displaystyle n!}
  
 times itself.</li>
<li>The alternatization of a <a href="/facts/Symmetric_function/IB8N4T3m">symmetric map</a> is zero.</li>
<li>The alternatization of a <a href="/facts/Bilinear_map/LH1lHVeP">bilinear map</a> is bilinear. Most notably, the alternatization of any <a href="/facts/Chain_complex/YAlchtCT">cocycle</a> is bilinear. This fact plays a crucial role in identifying the second <a href="/facts/Cohomology/J7FJzTNh">cohomology group</a> of a <a href="/facts/Lattice_(group)/2ls9Iv9N">lattice</a> with the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> of alternating <a href="/facts/Bilinear_form/gyvzn9ym">bilinear forms</a> on a lattice.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Alternating_algebra/kDoTSQXn">Alternating algebra</a></li>
<li><a href="/facts/Bilinear_map/LH1lHVeP">Bilinear map</a></li>
<li><a href="/facts/Exterior_algebra/rdN4TvpR">Exterior algebra § Alternating multilinear forms</a></li>
<li><a href="/facts/Map_(mathematics)/FABF1l2f">Map (mathematics)</a></li>
<li><a href="/facts/Multilinear_algebra/LrwpUFsM">Multilinear algebra</a></li>
<li><a href="/facts/Multilinear_map/YosSfBKE">Multilinear map</a></li>
<li><a href="/facts/Multilinear_form/Rk420AcK">Multilinear form</a></li>
<li><a href="/facts/Symmetrization/6XbIgHGx">Symmetrization</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki, N.</a> (2007). <i>Eléments de mathématique</i>. Vol. Algèbre Chapitres 1 à 3 (reprint ed.). Springer.</li>
<li>Dummit, David S.; Foote, Richard M. (2004). <i>Abstract Algebra</i> (3rd ed.). Wiley.</li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (2002). <i>Algebra</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 211 (revised 3rd ed.). Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-95385-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/48176673">48176673</a>.</li>
<li>Rotman, Joseph J. (1995). <i>An Introduction to the Theory of Groups</i>. Graduate Texts in Mathematics. Vol. 148 (4th ed.). Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-94285-8. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/30028913">30028913</a>.</li>
<li>Tu, Loring W. (2011). <i>An Introduction to Manifolds</i>. Springer-Verlag New York. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4419-7400-6.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Lang 2002, pp. 511–512 - Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211 (revised 3rd ed.). Springer. ISBN 978-0-387-95385-4. OCLC 48176673. <a href="https://search.worldcat.org/oclc/48176673" target="_blank">https://search.worldcat.org/oclc/48176673</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Bourbaki 2007, A III.80, §4 - Bourbaki, N. (2007). Eléments de mathématique. Vol. Algèbre Chapitres 1 à 3 (reprint ed.). Springer. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Lang 2002, pp. 511–512 - Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211 (revised 3rd ed.). Springer. ISBN 978-0-387-95385-4. OCLC 48176673. <a href="https://search.worldcat.org/oclc/48176673" target="_blank">https://search.worldcat.org/oclc/48176673</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Dummit & Foote 2004, p. 436 - Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Dummit & Foote 2004, p. 436 - Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Rotman 1995, p. 235 - Rotman, Joseph J. (1995). An Introduction to the Theory of Groups. Graduate Texts in Mathematics. Vol. 148 (4th ed.). Springer. ISBN 0-387-94285-8. OCLC 30028913. <a href="https://search.worldcat.org/oclc/30028913" target="_blank">https://search.worldcat.org/oclc/30028913</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Lang 2002, pp. 511–512 - Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211 (revised 3rd ed.). Springer. ISBN 978-0-387-95385-4. OCLC 48176673. <a href="https://search.worldcat.org/oclc/48176673" target="_blank">https://search.worldcat.org/oclc/48176673</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Tu 2011, p. 23 - Tu, Loring W. (2011). An Introduction to Manifolds. Springer-Verlag New York. ISBN 978-1-4419-7400-6. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

Alternating multilinear map open-in-new

Alternating multilinear map