Pairing

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For other uses, see pair.

The concept of pairing treated here occurs in mathematics.

[edit] Definition

Let R be a commutative ring with unity, and let M and N be two R-modules.

A pairing is any R-bilinear map e:M \times N \to R. That is, it satisfies

e(rm,n) = e(m,rn) = re(m,n)

for any r \in R. Or equivalently, a pairing is an R-linear map

M \otimes_R N \to R

where M \otimes_R N denotes the tensor product of M and N.

A pairing can also be considered as an R-linear map \Phi : M \to \operatorname{Hom}_{R} (N, R), which matches the first definition by setting Φ(m)(n): = e(m,n).

A pairing is called perfect if the above map Φ is an isomorphism of R-modules.

[edit] Examples

Any scalar product on a real vector space V is a pairing (set M = N = V, R = R in the above definitions).

The determinant map (2 × 2 matrices over k) → k can be seen as a pairing k^2 \times k^2 \to k.

[edit] Slightly different usages of the notion of pairing

Scalar products on complex vector spaces are sometimes called pairings, although they are not bilinear. For example, in representation theory, one has a scalar product on the characters of complex representations of a finite group which is frequently called character pairing.

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