Connection algebra

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 $C^{\mathrm{\infty }}(M)$-bimodule. Take the tensor algebra generated by this as a $C^{\mathrm{\infty }}(M)$-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 connection sheaf, which is a sheaf that associates to every open subset, the connection algebra over that open subset.