# Multilinear form

Jump to: navigation, search -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 .