Connection algebra

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This article gives a global construction for a differential manifold. There exists a sheaf analog of it, that associates a similar construct to every open subset. This sheaf analog is termed: sheaf of connection algebras

Definition

Let M be a differential manifold. The connection algebra of M, denoted \mathcal{C}(M), is defined as follows. Consider the Lie algebra of first-order differential operators on M, and treat it as a C^\infty(M)-bimodule. Take the tensor algebra generated by this as a C^\infty(M)-bimodule, and quotient it by the two-sided ideal generated by m(1) - 1. Here m(1) is the differential operator obtained as multiplication by the constant function 1.

The quotient algebra we get is termed the connection algebra on M.

The term connection algebra is also sometimes used for the sheaf of connection algebras, which is a sheaf that associates to every open subset, the connection algebra over that open subset.

References

Textbook references