Suppose we had a bilinear map V ⊗ W → C . What is meant by the fact that this map is a perfect pairing? How does one go about and show that something is a perfect pairing?

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Suppose we had a bilinear map   V  ⊗  W  →      C  . What is meant by the fact that this map is a perfect pairing? How does one go about and show that something is a perfect pairing?

Koen Harvey · Accepted Answer

B  :  V  ×  W  →  k be a k-bilinear map. (By the universal property of tensor products, this is the same thing as a k-linear map   V  ⊗  W  →  k.) Then, B induces two k-linear maps   L  :  V  →      W    ∗  ,   R  :  W  →      V    ∗   by partial evaluation:  L  (  v  )  =  (  w  ↦  B  (  v  ,  w  )  )  R  (  w  )  =  (  v  ↦  B  (  v  ,  w  )  )We say B is a perfect pairing if both L and R are isomorphisms. If V and W are both finite-dimensional, then this is equivalent to non-degeneracy of B. This means it suffices to check thatFor each v, if B(v,w)=0 for all w, then v=0.For each w, if B(v,w)=0 for all v, then w=0.

Suppose we had a bilinear map V⊗W→C. What is meant by the fact that this map is a perfect pairing? How does one go about and show that something is a perfect pairing?

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