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.