6
$\begingroup$

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.

$\endgroup$
5
$\begingroup$

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 $H^1(\mathfrak g,\mathfrak g)$)

Theoretically you can also identify the maps in the above exact sequence explicitly by chasing through the proof of the formality theorem.

$\endgroup$
2

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.