  Given a quantized analytic cycle $(X, \sigma)$ in $Y$, we give a categorical
Lie-theoretic interpretation of a geometric condition, discovered by Shilin Yu,
that involves the second formal neighbourhood of $X$ in $Y$. If this condition
(that we call tameness) is satisfied, we prove that the derived Ext algebra
$\mathcal{RH}om_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)$ is isomorphic to
the universal enveloping algebra of the shifted normal bundle
$\mathrm{N}_{X/Y}[-1]$ endowed with a specific Lie structure, strengthening an
earlier result of C\u{a}ld\u{a}raru, Tu, and the first author This approach
allows to get some conceptual proofs of many important results in the theory:
in the case of the diagonal embedding, we recover former results of Kapranov,
Markarian, and Ramadoss about (a) the Lie structure on the shifted tangent
bundle $\mathrm{T}_X[-1]$ (b) the corresponding universal enveloping algebra
(c) the calculation of Kapranov's big Chern classes. We also give a new
Lie-theoretic proof of Yu's result for the explicit calculation of the
quantized cycle class in the tame case: it is the Duflo element of the Lie
algebra object $\mathrm{N}_{X/Y}[-1]$.
