Multilinear form

$ 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)