Zoran Skoda symmetric ordering for Lie algebras

Given a Lie algebra $L$ over a field $k$ (for some facts it is enough to require just that $L$ is flat over the ground ring $k$), there are many vector space isomorphisms between $U(L)$ and $S(L)$; there is also a more narrow class of those vector space isomorphisms which are tautological on $k\oplus L$. A strong form of PBW theorem says that in characteristics zero the unique $k$-linear map $e: S(L)\to U(L)$ such that

$$y_1\cdots y_k \mapsto \frac{1}{k!}\sum_{\sigma\in S_k} \hat{y}_{\sigma 1} \cdots \hat{y}_{\sigma k}$$

for all $y_1\ldots, y_k\in L\subset S(L)$, where $\hat{y}\in L\subset U(L)$ is the same element as $y$ but understood as belonging to $U(L)$, is the isomorphism of $k$-coalgebras. This map is called the coexponential or the symmetrization map. It is also characterized by the property that for any element $y\in L$ its $n$-th power $y^n$ maps to $\hat{y}^n$. In fact the coexponential map is the only such isomorphism of coalgebras which is tautological on $k\oplus L$ and also functorial in $L$. Other isomorphism of coalgebras tautological on $k\oplus L$ are called generalized symmetrization maps.

Let $L$ be now $n$-dimensional over $L$ where $n$ is finite. There is an embedding of $U(L)$ into a semi-completed Weyl algebra $\hat{A}_n\cong S(L)\sharp \hat{S}(L^*)$ given by a universal formula on generators. Effectively, the elements of $U(L)$ are realized as “infinite order differential operators” (or first order if we use the automorphism $x_i\mapsto \partial^i, \partial_j\mapsto -x^j$. Suppose $x_1,\ldots,x_n$ to be a basis of $L$. Then the embedding is given on generators by $\hat{x}_\alpha \mapsto \sum_\lambda x_\lambda \phi^\lambda_\alpha$ where $\phi^\lambda_\alpha = \sum_{N=0}^\infty \frac{(-1)^N B_N }{N!}(\mathbb{C}^N)^\lambda_\alpha$, $B_N$ are Bernoulli numbers and $\mathbb{C}^N$ is the $N$-th power of the matrix $\mathbb{C}$ which is a $n\times n$-matrix with values in $L^*$ with entries $\mathbb{C}^\alpha_\beta = \sum_{\gamma = 1}^n C^\alpha_{\beta\gamma}\partial^\gamma$ and $C^\alpha_{\beta\gamma}$ are the structure constants determined by $[\hat{x}_\beta, \hat{x}_\gamma] = C^\alpha_{\beta\gamma} x^\alpha$.

If we compose the above embedding $U(L)\hookrightarrow \hat{A}_n$ with the action of $\hat{A}_n$ on the unit vector (“vacuum”) of the Fock module $S(L)$ we obtain the inverse of the symmetrization map.

The symmetrization map $e:S(L)\to U(L)$ transfers the noncommutative product from $U(L)$ to $S(L)$: if $f,g\in S(L)$ then $f\star g = e^{-1}(e(f)\cdot_{U(L)} e(g)$. The new product on $S(L)$ is called the star product in symmetric ordering.

• N. Bourbaki, Lie groups and algebras

• N. Durov, S. Meljanac, A. Samsarov, Z. Škoda, A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, Journal of Algebra 309, n.1, pp.318–359 (2007) math.RT/0604096.