Unimodular Form - Different Spaces

Different Spaces

Much of the theory is available for a bilinear mapping to the base field

B : V × W → F.

In this situation we still have induced linear mappings from V to W*, and from W to V*. It may happen that these mappings are isomorphisms; assuming finite dimensions, if one is an isomorphism, the other must be. When this occurs, B is said to be a perfect pairing.

In finite dimensions, this is equivalent to the pairing being nondegenerate (the spaces necessarily having the same dimensions). For modules (instead of vector spaces), just as how a nondegenerate form is weaker than a unimodular form, a nondegenerate pairing is a weaker notion than a perfect pairing. A pairing can be nondegenerate without being a perfect pairing, for instance Z × Z → Z via (x,y) ↦ 2xy is nondegenerate, but induces multiplication by 2 on the map Z → Z*.

Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product". To define them he uses diagonal matrices A_ij having only +1 or −1 for non-zero elements. Some of the "inner products" are symplectic forms and some are sesquilinear forms or Hermitian forms. Rather than a general field F, the instances with real numbers R, complex numbers C, and quaternions H are spelled out. The bilinear form

is called the real symmetric case and labeled R(p, q), where p + q = n. Then he articulates the connection to traditional terminology:

Some of the real symmetric cases are very important. The positive definite case R(n, 0) is called Euclidean space, while the case of a single minus, R(n−1, 1) is called Lorentzian space. If n = 4, then Lorentzian space is also called Minkowski space or Minkowski spacetime. The special case R(p, p) will be referred to as the split-case.