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)