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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m0652902.png" />-linear form, on a unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m0652904.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m0652905.png" />''
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A [[Multilinear mapping|multilinear mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m0652906.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m0652907.png" /> is a commutative associative ring with a unit, cf. [[Associative rings and algebras|Associative rings and algebras]]). A multilinear form is also called a multilinear function (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m0652909.png" />-linear function). Since a multilinear form is a particular case of a multilinear mapping, one can speak of symmetric, skew-symmetric, alternating, symmetrized, and skew-symmetrized multilinear forms. For example, the [[Determinant|determinant]] of a square [[Matrix|matrix]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529010.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529011.png" /> is a skew-symmetrized (and therefore alternating) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529012.png" />-linear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529013.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529014.png" />-linear forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529015.png" /> form an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529016.png" /> module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529017.png" />, which is naturally isomorphic to the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529018.png" /> of all linear forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529019.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529020.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529021.png" />), one speaks of bilinear forms (cf. [[Bilinear form|Bilinear form]]) (respectively, trilinear forms).
+
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 +
{{TEX|done}}
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529022.png" />-linear forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529023.png" /> are closely related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529024.png" />-times covariant tensors, i.e. elements of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529025.png" />. More precisely, there is a linear mapping
+
'' $  n $-
 +
linear form, on a unitary  $  A $-
 +
module $  E $''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529026.png" /></td> </tr></table>
+
A [[Multilinear mapping|multilinear mapping]]  $  E  ^ {n} \rightarrow A $(
 +
here  $  A $
 +
is a commutative associative ring with a unit, cf. [[Associative rings and algebras|Associative rings and algebras]]). A multilinear form is also called a multilinear function ( $  n $-
 +
linear function). Since a multilinear form is a particular case of a multilinear mapping, one can speak of symmetric, skew-symmetric, alternating, symmetrized, and skew-symmetrized multilinear forms. For example, the [[Determinant|determinant]] of a square [[Matrix|matrix]] of order  $  n $
 +
over  $  A $
 +
is a skew-symmetrized (and therefore alternating)  $  n $-
 +
linear form on  $  A  ^ {n} $.
 +
The  $  n $-
 +
linear forms on  $  E $
 +
form an  $  A $
 +
module  $  L _ {n} ( E, A) $,
 +
which is naturally isomorphic to the module  $  (\otimes  ^ {n} E)  ^ {*} $
 +
of all linear forms on  $  \otimes  ^ {n} E $.
 +
In the case  $  n = 2 $(
 +
$  n = 3 $),
 +
one speaks of bilinear forms (cf. [[Bilinear form|Bilinear form]]) (respectively, trilinear forms).
 +
 
 +
The  $  n $-
 +
linear forms on  $  E $
 +
are closely related to  $  n $-
 +
times covariant tensors, i.e. elements of the module  $  T  ^ {n} ( E  ^ {*} ) = \otimes  ^ {n} E  ^ {*} $.  
 +
More precisely, there is a linear mapping
 +
 
 +
$$
 +
\gamma _ {n} :  T  ^ {n} ( E  ^ {*} )  \rightarrow  L _ {n} ( E, A),
 +
$$
  
 
such that
 
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529027.png" /></td> </tr></table>
+
$$
 +
\gamma _ {n} ( u _ {1} \otimes \dots \otimes u _ {n} )( x _ {1} \dots
 +
x _ {n} )  = u _ {1} ( x _ {1} ) \dots u _ {n} ( x _ {n} )
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529029.png" />. If the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529030.png" /> is free (cf. [[Free module|Free module]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529031.png" /> is injective, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529032.png" /> is also finitely generated, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529033.png" /> is bijective. In particular, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529034.png" />-linear forms on a finite-dimensional vector space over a field are identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529035.png" />-times covariant tensors.
+
for any $  u _ {i} \in E  ^ {*} $,  
 +
$  x _ {i} \in E $.  
 +
If the module $  E $
 +
is free (cf. [[Free module|Free module]]), $  \gamma $
 +
is injective, while if $  E $
 +
is also finitely generated, $  \gamma $
 +
is bijective. In particular, the $  n $-
 +
linear forms on a finite-dimensional vector space over a field are identified with $  n $-
 +
times covariant tensors.
  
For any forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529037.png" /> one can define the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529038.png" /> via the formula
+
For any forms $  u \in L _ {n} ( E, A) $,
 +
$  v \in L _ {m} ( E, A) $
 +
one can define the tensor product $  u \otimes v \in L _ {n+m} ( E, A) $
 +
via the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529039.png" /></td> </tr></table>
+
$$
 +
u \otimes v ( x _ {1} \dots x _ {n+m} )  = \
 +
u( x _ {1} \dots x _ {n} ) v( x _ {n+1} \dots x _ {n+m} ).
 +
$$
  
For symmetrized multilinear forms (cf. [[Multilinear mapping|Multilinear mapping]]), a symmetrical product is also defined:
+
For symmetrized multilinear forms (cf. [[Multilinear mapping]]), a symmetrical product is also defined:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529040.png" /></td> </tr></table>
+
$$
 +
( \sigma _ {n} u) \lor ( \sigma _ {m} v)  = \sigma _ {n+m} ( u \otimes v),
 +
$$
  
 
while for skew-symmetrized multilinear forms there is an exterior product
 
while for skew-symmetrized multilinear forms there is an exterior product
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529041.png" /></td> </tr></table>
+
$$
 +
( \alpha _ {n} u) \wedge ( \alpha _ {m} v)  = \
 +
\alpha _ {n+m} ( u \otimes v).
 +
$$
  
These operations are extended to the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529044.png" />, to the module of symmetrized forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529045.png" /> and to the module of skew-symmetrized forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529046.png" /> respectively, which transforms them into associative algebras with a unit. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529047.png" /> is a finitely-generated free module, then the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529048.png" /> define an isomorphism of the [[Tensor algebra|tensor algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529050.png" /> and the [[Exterior algebra|exterior algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529051.png" /> on the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529052.png" />, which in that case coincides with the algebra of alternating forms. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529053.png" /> is a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529054.png" />, then there is also an isomorphism of the symmetric algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529055.png" /> on the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529056.png" /> of symmetric forms.
+
These operations are extended to the module $  L _  \star  ( E, A) = \oplus_{n=0}  ^  \infty  L( E, A) $,  
 +
where $  L _ {0} ( E, A) = A $,
 +
$  L _ {1} ( E, A) = E  ^ {*} $,  
 +
to the module of symmetrized forms $  L _  \sigma  ( E, A) = \oplus_{n=0} ^  \infty  \sigma _ {n} L _ {n} ( E, A) $
 +
and to the module of skew-symmetrized forms $  L _  \alpha  ( E, A) = \oplus_{n=0} ^  \infty  \alpha _ {n} L _ {n} ( E, A) $
 +
respectively, which transforms them into associative algebras with a unit. If $  E $
 +
is a finitely-generated free module, then the mappings $  \gamma _ {n} $
 +
define an isomorphism of the [[Tensor algebra|tensor algebra]] $  T( E  ^ {*} ) $
 +
on $  L _  \star  ( E, A) $
 +
and the [[Exterior algebra|exterior algebra]] $  \Lambda ( E  ^ {*} ) $
 +
on the algebra $  L _  \alpha  ( E, A) $,  
 +
which in that case coincides with the algebra of alternating forms. If $  A $
 +
is a field of characteristic 0 $,  
 +
then there is also an isomorphism of the symmetric algebra $  S( E  ^ {*} ) $
 +
on the algebra $  L _  \sigma  ( E, A) $
 +
of symmetric forms.
  
Any multilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529057.png" /> corresponds to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529058.png" />, given by the formula
+
Any multilinear form $  u \in L _ {n} ( E, A) $
 +
corresponds to a function $  \omega _ {n} ( u): E \rightarrow A $,  
 +
given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529059.png" /></td> </tr></table>
+
$$
 +
\omega _ {n} ( u)( x)  = u( x \dots x),\  x \in E.
 +
$$
  
Functions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529060.png" /> are called forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529062.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529063.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529064.png" /> is a free module, then in coordinates relative to an arbitrary basis they are given by homogeneous polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529065.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529066.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529067.png" />) one obtains quadratic (cubic) forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529068.png" /> (cf. [[Quadratic form|Quadratic form]]; [[Cubic form|Cubic form]]). The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529069.png" /> completely determines the symmetrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529070.png" /> of a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529071.png" />:
+
Functions of the form $  \omega _ {n} ( u) $
 +
are called forms of degree $  n $
 +
on $  E $;  
 +
if $  E $
 +
is a free module, then in coordinates relative to an arbitrary basis they are given by homogeneous polynomials of degree $  n $.  
 +
In the case $  n = 2 $(
 +
$  n= 3 $)  
 +
one obtains quadratic (cubic) forms on $  E $(
 +
cf. [[Quadratic form|Quadratic form]]; [[Cubic form|Cubic form]]). The form $  F = \omega ( u) $
 +
completely determines the symmetrization $  \sigma _ {n} u $
 +
of a form $  u \in L _ {n} ( E, A) $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529072.png" /></td> </tr></table>
+
$$
 +
\sigma _ {n} u( x _ {1} \dots x _ {n} ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529073.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {r=1} ^ { n }  (- 1)  ^ {n-r} \sum _ {i _ {1} < \dots < i _ {r} } F( x _ {i _ {1}  } + \dots + x _ {i _ {r}  } ).
 +
$$
  
In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529074.png" />,
+
In particular, for $  n= 2 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529075.png" /></td> </tr></table>
+
$$
 +
( \sigma _ {2} u)( x, y)  = \
 +
F( x+ y) - F( x) - F( y).
 +
$$
  
The mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529077.png" /> define a homomorphism of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529078.png" /> on the algebra of all polynomial functions (cf. [[Polynomial function|Polynomial function]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529079.png" />, which is an isomorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529080.png" /> is a finitely-generated free module over an infinite integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065290/m06529081.png" />.
+
The mappings $  \gamma _ {n} $
 +
and $  \omega _ {n} $
 +
define a homomorphism of the algebra $  S( E  ^ {*} ) $
 +
on the algebra of all polynomial functions (cf. [[Polynomial function|Polynomial function]]) $  P( E) $,  
 +
which is an isomorphism if $  E $
 +
is a finitely-generated free module over an infinite integral domain $  A $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley  (1974)  pp. Chapt.1;2  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley  (1975)  pp. Chapt.4;5;6  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1984)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley  (1974)  pp. Chapt.1;2  (Translated from French)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley  (1975)  pp. Chapt.4;5;6  (Translated from French)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1984)</TD></TR>
 +
</table>

Latest revision as of 20:46, 16 January 2024


$ n $- linear form, on a unitary $ A $- module $ E $

A multilinear mapping $ E ^ {n} \rightarrow A $( here $ A $ is a commutative associative ring with a unit, cf. Associative rings and algebras). A multilinear form is also called a multilinear function ( $ n $- linear function). Since a multilinear form is a particular case of a multilinear mapping, one can speak of symmetric, skew-symmetric, alternating, symmetrized, and skew-symmetrized multilinear forms. For example, the determinant of a square matrix of order $ n $ over $ A $ is a skew-symmetrized (and therefore alternating) $ n $- linear form on $ A ^ {n} $. The $ n $- linear forms on $ E $ form an $ A $ module $ L _ {n} ( E, A) $, which is naturally isomorphic to the module $ (\otimes ^ {n} E) ^ {*} $ of all linear forms on $ \otimes ^ {n} E $. In the case $ n = 2 $( $ n = 3 $), one speaks of bilinear forms (cf. Bilinear form) (respectively, trilinear forms).

The $ n $- linear forms on $ E $ are closely related to $ n $- times covariant tensors, i.e. elements of the module $ T ^ {n} ( E ^ {*} ) = \otimes ^ {n} E ^ {*} $. More precisely, there is a linear mapping

$$ \gamma _ {n} : T ^ {n} ( E ^ {*} ) \rightarrow L _ {n} ( E, A), $$

such that

$$ \gamma _ {n} ( u _ {1} \otimes \dots \otimes u _ {n} )( x _ {1} \dots x _ {n} ) = u _ {1} ( x _ {1} ) \dots u _ {n} ( x _ {n} ) $$

for any $ u _ {i} \in E ^ {*} $, $ x _ {i} \in E $. If the module $ E $ is free (cf. Free module), $ \gamma $ is injective, while if $ E $ is also finitely generated, $ \gamma $ is bijective. In particular, the $ n $- linear forms on a finite-dimensional vector space over a field are identified with $ n $- times covariant tensors.

For any forms $ u \in L _ {n} ( E, A) $, $ v \in L _ {m} ( E, A) $ one can define the tensor product $ u \otimes v \in L _ {n+m} ( E, A) $ via the formula

$$ u \otimes v ( x _ {1} \dots x _ {n+m} ) = \ u( x _ {1} \dots x _ {n} ) v( x _ {n+1} \dots x _ {n+m} ). $$

For symmetrized multilinear forms (cf. Multilinear mapping), a symmetrical product is also defined:

$$ ( \sigma _ {n} u) \lor ( \sigma _ {m} v) = \sigma _ {n+m} ( u \otimes v), $$

while for skew-symmetrized multilinear forms there is an exterior product

$$ ( \alpha _ {n} u) \wedge ( \alpha _ {m} v) = \ \alpha _ {n+m} ( u \otimes v). $$

These operations are extended to the module $ L _ \star ( E, A) = \oplus_{n=0} ^ \infty L( E, A) $, where $ L _ {0} ( E, A) = A $, $ L _ {1} ( E, A) = E ^ {*} $, to the module of symmetrized forms $ L _ \sigma ( E, A) = \oplus_{n=0} ^ \infty \sigma _ {n} L _ {n} ( E, A) $ and to the module of skew-symmetrized forms $ L _ \alpha ( E, A) = \oplus_{n=0} ^ \infty \alpha _ {n} L _ {n} ( E, A) $ respectively, which transforms them into associative algebras with a unit. If $ E $ is a finitely-generated free module, then the mappings $ \gamma _ {n} $ define an isomorphism of the tensor algebra $ T( E ^ {*} ) $ on $ L _ \star ( E, A) $ and the exterior algebra $ \Lambda ( E ^ {*} ) $ on the algebra $ L _ \alpha ( E, A) $, which in that case coincides with the algebra of alternating forms. If $ A $ is a field of characteristic $ 0 $, then there is also an isomorphism of the symmetric algebra $ S( E ^ {*} ) $ on the algebra $ L _ \sigma ( E, A) $ of symmetric forms.

Any multilinear form $ u \in L _ {n} ( E, A) $ corresponds to a function $ \omega _ {n} ( u): E \rightarrow A $, given by the formula

$$ \omega _ {n} ( u)( x) = u( x \dots x),\ x \in E. $$

Functions of the form $ \omega _ {n} ( u) $ are called forms of degree $ n $ on $ E $; if $ E $ is a free module, then in coordinates relative to an arbitrary basis they are given by homogeneous polynomials of degree $ n $. In the case $ n = 2 $( $ n= 3 $) one obtains quadratic (cubic) forms on $ E $( cf. Quadratic form; Cubic form). The form $ F = \omega ( u) $ completely determines the symmetrization $ \sigma _ {n} u $ of a form $ u \in L _ {n} ( E, A) $:

$$ \sigma _ {n} u( x _ {1} \dots x _ {n} ) = $$

$$ = \ \sum _ {r=1} ^ { n } (- 1) ^ {n-r} \sum _ {i _ {1} < \dots < i _ {r} } F( x _ {i _ {1} } + \dots + x _ {i _ {r} } ). $$

In particular, for $ n= 2 $,

$$ ( \sigma _ {2} u)( x, y) = \ F( x+ y) - F( x) - F( y). $$

The mappings $ \gamma _ {n} $ and $ \omega _ {n} $ define a homomorphism of the algebra $ S( E ^ {*} ) $ on the algebra of all polynomial functions (cf. Polynomial function) $ P( E) $, which is an isomorphism if $ E $ is a finitely-generated free module over an infinite integral domain $ A $.

References

[1] N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)
[2] N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)
[3] S. Lang, "Algebra" , Addison-Wesley (1984)
How to Cite This Entry:
Multilinear form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multilinear_form&oldid=16677
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article