Differentials on an Algebra

Differentiating the State Evaluation Map from Matrices to Functions on Projective Space

“…A theory of how to do this is set down in [3], (using left instead of right connections), but it assumes conditions on the curvature that we do not have and results in a cochain map, so we need to be more careful and give a more general account of the theory, beginning with how σ E extends to a map of differential forms, with general algebras A, B, and bimodule W.…”

“…We begin with a right handed version of Lemma 3.72 in [3]. For algebras A, B with calculi, we suppose that (∇ W , σ W ) is a bimodule connection on an A-B bimodule W. The curvature R W of a right bimodule connection must be a right module map but not necessarily a bimodule map.…”

“…Under the more restrictive conditions where R W is a bimodule map [3] φ would be a cochain map. However, more generally we find a correction term.…”

“…In Proposition 1, we see that under the condition of Lemma 1 there is a functor ⊗ W from E A to E B , using the specified connection on the tensor product. We would like to calculate the curvature of this tensor product connection, but as we noted before the curvature of W is not necessarily a left module map, so we need more generality than in [3].…”

“…Then, we use the methods of connections on bimodules to connect the differential structure on M n (C) (we take the universal calculus) to that on CP n−1 (the usual calculus). Here, we follow the methods in [3] but then find that the conditions required there do not apply, so in Section 5.1 we consider a more general theory extending the results in [3]. As a result, Proposition 12 on an induced functor from left M n -modules to holomorphic bundles on CP n−1 is phrased in terms of holomorphic bundles rather than flat bundles on CP n−1 .…”