Pairing

The concept of pairing treated here occurs in mathematics.

= 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 imes N o 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 o 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 o operatorname{Hom}_{R} (N, R) , which matches the first definition by setting Phi (m) (n) := e(m,n) .

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

= 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 imes k^2 o k.

= 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.