Multiplication in universal enveloping algebra in terms of PBW isomorphism

Question

Let $\mathfrak g$ be a Lie algebra. Consider the multiplication map $m:\mathfrak g\otimes U(\mathfrak g)\to U(\mathfrak g)$ and $i:S(\mathfrak g)\to U(\mathfrak g)$ -- Poincare-Birkhoff-Witt isomorphism. Is there an explicit formula for $$i^{-1}\circ m\circ id\otimes i:\mathfrak{g}\otimes S(\mathfrak g)\to S(\mathfrak g)$$ in terms of, say, multiplication in $S(\mathfrak g)$ and its $\mathfrak g$-module structure?

Answer 1

$\newcommand{\g}{\mathfrak g}$ There are several (equivalent) answers to this question.

1. Let $BCH$ be the Baker-Campbell-Hausdorff formula and write $$BCH(x,y)=\sum_{p,q\geq 0} BCH_{p,q}(x,y)$$ where $BCH_{p,q}(x,y)$ is the homogenous part of $x$ degree $p$ and $y$ degree $q$ in the Free Lie algebra on $x,y$. Then for any $x,y \in \g$, the product $x^py^q$ is given by: $$\sum_{k\geq 0}\frac{p!q!}{k!} \sum_{(p_1,\dots,p_k)\in Part_k(p),(q_1,\dots,q_k)\in Part_k(q)} i(BCH_{p_1,q_1}(x,y)\dots BCH_{p_k,q_k}(x,y))$$ Note that for $p=1$ (which is the case you are interested in) there are well known explicit formulas for $BCH_{1,q}$ in terms of Bernouilli numbers, see e.g. equation 2.22 in http://arxiv.org/abs/math/9905080
2. The "tree part" of Kontsevich star product coincide with the PBW multiplication. An answer to your question using this formalism is in section 3 of http://arxiv.org/abs/math/9905080. Again you get an explicit formula involving Bernouilli numbers.
3. This product is also known as the 'Gutt star product" in the litterature. Indeed, Gutt showed that there is a fairly explicit star product on $T^*G$ whic in turn can be used to compute the PBW star product on $S(\g) $(http://link.springer.com/article/10.1007/BF00400441)

All those formulas are thought of quantization of the Poisson manifold $\g^*$ where $\g$ is a finite-dimensional Lie algebra. However, at the end of the day the formulas make sense in the infinite-dimensional case as well (which would not be the case, e.g., if you did not throw away the wheels in 2).

Answer 2

You may want to read Equation (27) and Proposition 5.10 in the paper https://arxiv.org/pdf/1408.2903.pdf Your question is the special case $L=\mathfrak{g}$, $A=0$.