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I am interested in finding formulas for commutators of symmetrized monomials in a universal enveloping algebra. Let $C(x_1,\ldots, x_n)= (1/n!)\sum x_{\sigma(1)}\cdots x_{\sigma(n)}$ where the sum runs over all permutations, and $x_i \in L$ for some LIe algebra $L$. This is an element of $UL$.

Now, reasonable combinatorics shows that, in $UL$, we have,

$ [C(x_1,\ldots, x_n), l] = \sum_{i=1}^n C(\ldots,[x_i,l],\ldots) $

for $l\in L$.

I am looking for formulas for $[C(x_1,\ldots, x_n), C(y_1,\ldots, y_m)]$ in terms of symmetrized monomials and brackets. Even for $n=m=2$ the number of terms gets fairly large. If anyone knows where I can find such things I would be very grateful.

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What is exact formulas for the case $n=m=2$? –  Melania Feb 8 '11 at 6:54
    
Presumably you are working over an arbitrary field of characteristic 0 here. As Melania comments, one should try to answer the question first for products of length 2. Anyway I can't visualize an illuminating general answer, since the symmetrization process already depends so heavily on the specific nature of the bracket in the Lie algebra. Maybe in very special cases a good formula could be written down? –  Jim Humphreys Feb 8 '11 at 14:09
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1 Answer 1

This is probably not yet a final answer but may shine some additional light on the problem: For simplicity, I assume that $L$ is finite-dimensional and defined over the reals (for some other field of char $0$, the following should still work).

The symmetrization map can be viewed as a linear map \begin{equation} \sigma\colon \mathrm{S}(L) \longrightarrow \mathrm{U}(L) \end{equation} from the symmetric algebra over $L$ into the universal envelopping algebra. It is now possible (essentially via PBW) to show that this is a fitration compatible linear bijection. Thus it allows to pull-back the product of $\mathrm{U}(L)$ to $\mathrm{S}(L)$. The result is the star product of Gutt / Drinfel'd (both in 1983, I guess). A further canonical isomorphism yields that the symmetric algebra is nothing else than the poylnomials on the dual $L^*$ (suppose $L$ is finite-dimensional for convenience) Thus your question is equivalent to the following task:

What is the Gutt star product commutator of two (homogeneous) polynomials on $L^*$?

Gutt has computed many properties of this star product and fan almost explicit formula. However, it essentially involved the full BCH series of $L$, so my guess is that a complete answer might be as complicated as computing BCH.

The Gutt star product can be characterized nicely as follows: take $x, y \in L$ and view them as linear polynomials on $L^*$ as usual. Then form the formal exponential functions $e_{\hbar x} (\alpha) = \exp(\hbar \alpha(x))$ and similarly for $e_{\hbar y}$ where $\hbar$ is your formal parameter. Then $\star_{\mathrm{Gutt}}$ is uniquely determined by \begin{equation} e_{\hbar x} \star_{\mathrm{Gutt}} e_{\hbar y} = e_{\mathrm{BHC}(\hbar x, \hbar y)} \end{equation} I hope I got the signs right :) You can find this formulas also in Section 8 of q-alg/9707030 (published in Commun. Math. Phys.). There are many more papers on the Gutt star product, so a little MRsearch will probably give some addition info.

The solution of your problem is now obtained by differentiating the above equation with respect to $\hbar$ sufficiently often and use polarization afterwarts. But as I sad, you need to know BCH quite well to efficiently do that. In the end you take commutators\ldots

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