# Connection algebra

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 $M$.