  Let $X$ be a smooth scheme over a field of characteristic 0. Let
$\dd^{\bullet}(X)$ be the complex of polydifferential operators on $X$ equipped
with Hochschild co-boundary. Let $L(\dd^1(X))$ be the free Lie algebra
generated over $\strc$ by $\dd^1(X)$ concentrated in degree 1 equipped with
Hochschild co-boundary. We have a symmetrization map $I: \oplus_k
\sss^k(L(\dd^1(X))) \rar \dd^{\bullet}(X)$. Theorem 1 of this paper measures
how the map $I$ fails to commute with multiplication.
  A consequence of Theorem 1 and Theorem 2 is Corollary 1, a result "dual" to
Theorem 1 of Markarian [3] that measures how the Hochschild-Kostant-Rosenberg
quasi-isomorphism fails to commute with multiplication.
  In order to understand Theorem 1 conceptually, we prove a theorem (Theorem 3)
stating that $\dd^{\bullet}(X)$ is the universal enveloping algebra of
$T_X[-1]$ in $\dcat$. An easy consequence of Theorem 3 is Theorem 4, which
interprets the Chern character $E$ as the "character of the representation $E$
of $T_X[-1]$" and gives a description of the big Chern classes of $E$.
  Finally, Theorem 4 along with Theorem 1 is used to give an explicit formula
(Theorem 5) expressing the big Chern classes of $E$ in terms of the components
of the Chern character of $E$.
