  We prove a number of results on integrability and extendability of Lie
algebras of unbounded skew-symmetric operators with common dense domain in
Hilbert space. By integrability for a Lie algebra $\mathfrak{g}$, we mean that
there is an associated unitary representation $\mathcal{U}$ of the
corresponding simply connected Lie group such that $\mathfrak{g}$ is the
differential of $\mathcal{U}$. Our results extend earlier integrability results
in the literature; and are new even in the case of a single operator. Our
applications include a new invariant for certain Riemann surfaces.
