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Pairing
From Wikipedia, the free encyclopedia
The concept of pairing treated here occurs in mathematics.
Let R be a commutative ring with unity, and let M and N be two R-modules.
A pairing is any R-bilinear map 
. That is, it satisfies
- e(rm,n) = e(m,rn) = re(m,n)
for any 
. Or equivalently, a pairing is an R-linear map
where 
denotes the tensor product of M and N.
A pairing can also be considered as an R-linear map 
, 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.
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 
.
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.