Multilinear mapping

$n$- linear mapping, multilinear operator

A mapping $f$ of the direct product $\prod _{i=1} ^ {n} E _ {i}$ of unitary modules $E _ {i}$( cf. Unitary module) over a commutative associative ring $A$ with a unit into a certain $A$- module $F$ which is linear in each argument, i.e. which satisfies the condition

$$f( x _ {1} \dots x _{i-1} , ay + bz, x _ {i+1} \dots x _ {n} ) =$$

$$= \ af( x _ {1} \dots x _{i-1} , y, x _ {i+1} \dots x _ {n} ) +$$

$$+ bf ( x _ {i} \dots x _{i-1} , z , x _ {i+1} \dots x _ {n} )$$

$$( a, b \in A; \ y, z \in E _ {i} ,\ i = 1 \dots n).$$

In the case $n= 2$( $n= 3$) one speaks of a bilinear mapping (respectively, a trilinear mapping). Each multilinear mapping

$$f: \prod _ {i=1} ^ { n } E _ {i} \rightarrow F$$

defines a unique linear mapping $\overline{f}\;$ of the tensor product $\otimes _{i=1} ^ {n} E _ {i}$ into $F$ such that

$$\overline{f}\; ( x _ {1} \otimes \dots \otimes x _ {n} ) = \ f( x _ {1} \dots x _ {n} ),\ x _ {i} \in E _ {i} ,$$

where the correspondence $f \mapsto \overline{f}\;$ is a bijection of the set of multilinear mappings $\prod _{i=1} ^ {n} E _ {i} \rightarrow F$ into the set of all linear mappings $\otimes _{i=1} ^ {n} E _ {i} \rightarrow F$. The multilinear mappings $\prod _{i=1} ^ {n} E _ {i} \rightarrow F$ naturally form an $A$- module.

On the $A$- module $L _ {n} ( E, F )$ of all $n$- linear mappings $E ^ {n} \rightarrow F$ there acts the symmetric group $S _ {n}$:

$$( sf )( x _ {1} \dots x _ {n} ) = \ f( x _ {s(} 1) \dots x _ {s(} n) ),$$

where $s \in S _ {n}$, $f \in L _ {n} ( E, F )$, $x _ {i} \in E$. A multilinear mapping $f$ is called symmetric if $sf = f$ for all $s \in S _ {n}$, and skew-symmetric if $sf = \epsilon ( s) f$, where $\epsilon ( s) = \pm 1$ in accordance with the sign of the permutation $s$. A multilinear mapping is called sign-varying (or alternating) if $f( x _ {1} \dots x _ {n} ) = 0$ when $x _ {i} = x _ {j}$ for some $i \neq j$. Any alternating multilinear mapping is skew-symmetric, while if in $F$ the equation $2y = 0$ has the unique solution $y = 0$ the converse also holds. The symmetric multilinear mappings form a submodule in $L _ {n} ( E, F )$ that is naturally isomorphic to the module of linear mappings $L( S ^ {n} E, F )$, where $S ^ {n} E$ is the $n$- th symmetric power of $E$( see Symmetric algebra). The alternating multilinear mappings form a submodule that is naturally isomorphic to $L( \Lambda ^ {n} E, F )$, where $\Lambda ^ {n} E$ is the $n$- th exterior power of the module $E$( see Exterior algebra). The multilinear mapping $\alpha _ {n} f = \sum _ {s \in S _ {n} } sf$ is called the symmetrized multilinear mapping defined by $f$, while the multilinear mapping $\sigma _ {n} f = \sum _ {s \in S _ {n} } \epsilon ( s) sf$ is called the skew-symmetrized mapping defined by $f$. Symmetrized (skew-symmetrized) multilinear mappings are symmetric (respectively, alternating), and if in $F$ the equation $n!y = c$ has a unique solution for each $c \in F$, then the converse is true. A sufficient condition for any alternating multilinear mapping to be a skew-symmetrization is that the module $E$ is free (cf. Free module). For references see Multilinear form.