Multilinear form

-linear form, on a unitary -module

A multilinear mapping (here is a commutative associative ring with a unit, cf. Associative rings and algebras). A multilinear form is also called a multilinear function (-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 over is a skew-symmetrized (and therefore alternating) -linear form on . The -linear forms on form an module , which is naturally isomorphic to the module of all linear forms on . In the case (), one speaks of bilinear forms (cf. Bilinear form) (respectively, trilinear forms).

The -linear forms on are closely related to -times covariant tensors, i.e. elements of the module . More precisely, there is a linear mapping

such that

for any , . If the module is free (cf. Free module), is injective, while if is also finitely generated, is bijective. In particular, the -linear forms on a finite-dimensional vector space over a field are identified with -times covariant tensors.

For any forms , one can define the tensor product via the formula

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

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

These operations are extended to the module , where , , to the module of symmetrized forms and to the module of skew-symmetrized forms respectively, which transforms them into associative algebras with a unit. If is a finitely-generated free module, then the mappings define an isomorphism of the tensor algebra on and the exterior algebra on the algebra , which in that case coincides with the algebra of alternating forms. If is a field of characteristic , then there is also an isomorphism of the symmetric algebra on the algebra of symmetric forms.

Any multilinear form corresponds to a function , given by the formula

Functions of the form are called forms of degree on ; if is a free module, then in coordinates relative to an arbitrary basis they are given by homogeneous polynomials of degree . In the case () one obtains quadratic (cubic) forms on (cf. Quadratic form; Cubic form). The form completely determines the symmetrization of a form :

In particular, for ,

The mappings and define a homomorphism of the algebra on the algebra of all polynomial functions (cf. Polynomial function) , which is an isomorphism if is a finitely-generated free module over an infinite integral domain .

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)