# Derivations of universal enveloping algebra of Lie algebras

We know a lot about derivations of Lie algebra. However, for the universal enveloping algebra of Lie algebra, we have few references about it. My question: describing the derivations of enveloping algebras is very hard or easy? And are there any applications about it?

For example, for the one-dimensional Lie algebra, the derivations of the universal enveloping algebra form the one-sided Witt algebra.

Associated to any associative algebra $$A$$ is the Hochschild cochain complex \begin{align*} HH^n(A) &= \operatorname{Hom}(A^{\otimes n},A),\\ \mathrm d f(a_0,\dots,a_n) &= a_0f(a_1,\dots,a_n) + (-1)^{n+1} f(a_0,\dots,a_{n-1})a_n + \sum_{i=1}^n (-1)^i f(a_0,\dots,a_{i-1}a_i,\dots,a_n) \end{align*} In particular, $$\operatorname{ker} d^1\subset \operatorname{Hom}(A,A)$$ are exactly the derivations, and $$\operatorname{im} d^0\subset \operatorname{Hom}(A,A)$$ are the inner derivations. Thus there is an exact sequence $$0\to Z(A)= HH^0(A)\to A\to \operatorname{Der}(A)\to HH^1(A)\to 0$$ For $$A = \operatorname{Sym}(V)$$, we have $$HH^i(A)\cong \operatorname{Hom}(\Lambda^i V,\operatorname{Sym}(V))$$ (the Hochschild-Kostant-Rosenberg isomorphism). In this case this exact sequence encodes that a derivation is uniquely and arbitrarily defined by what it does on generators. This cochain complex carries an obvious commutative multiplication and a $$1$$-shifted Lie bracket which uniquely extends the commutator of derivations and their action on algebra elements as a biderivation.
For $$A = U\mathfrak g$$, the PBW filtration defines a filtration on the Hochschild complex, and the associated graded is $$HH^*(\operatorname{Sym}(\mathfrak g))$$. Thus you get a spectral sequence with $$E^1$$-page $$\operatorname{Hom}(\Lambda^i \mathfrak g,\operatorname{Sym}(\mathfrak g))$$, where the differential is given by taking the Lie bracket $$[\Pi,-]$$ with the element $$\Pi\in\operatorname{Hom}(\Lambda^2\mathfrak g,\mathfrak g)$$ defining the Lie structure (the fact that this squares to zero is equivalent to the Jacobi identity). In fact, this complex can be identified with the Chevalley-Eilenberg complex calculating the Lie algebra cohomology of $$\mathfrak g$$ with values in $$\operatorname{Sym}(\mathfrak g)$$. A quite deep theorem by Kontsevich implies that this spectral sequence degenerates at the $$E^2$$-page (compare Pevzner, Michaël, and Ch Torossian. "Isomorphisme de Duflo et la cohomologie tangentielle." Journal of geometry and Physics 51.4 (2004): 486-505.). This implies the following for the two groups we are interested in:
• $$Z(U\mathfrak g) = HH^0(U\mathfrak g)\cong (\operatorname{Sym}\mathfrak g)^{\mathfrak g}$$ (Duflo isomorphism)
• $$HH^1(U\mathfrak g)\cong H^1(\mathfrak g,\operatorname{Sym}\mathfrak g)$$ (recall that Lie derivations are given by $$H^1(\mathfrak g,\mathfrak g)$$)