Let $\frak{g}$ be a semisimple Lie algebra and $U(\frak{g})$ its universal enveloping algebra. The adjoint action of $\frak{g}$ on itself extends to an action of $\frak{g}$ on $U(\frak{g})$. How does this action interact with the PBW basis of $U(\frak{g})$? More precisely: are the irreducible submodules of $U(\frak{g})$ spanned by the elements of the PBW basis?



Adjoint action on the universal enveloping algebra and the PBW theorem
Since the sum of all the trivial subrepresentations of $U(\mathfrak{g})$ with the adjoint action is its centre, if you could do this then the centre must have a monomial basis but it does not.


For example even if $\mathfrak{g}=\mathfrak{sl}\_2$ with basis $e,h,f$ in that order then the centre consists of polynomials in $$C=ef+fe+\frac 1 2 h^2=2ef+\frac 1 2(h^2-h)$$ which does not live in the monomial basis.


