Commutator formulas in a universal enveloping algebra - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:19:02Z http://mathoverflow.net/feeds/question/54723 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54723/commutator-formulas-in-a-universal-enveloping-algebra Commutator formulas in a universal enveloping algebra Justin Young 2011-02-08T03:44:58Z 2011-02-08T15:34:23Z <p>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 $UL$. </p> <p>Now, reasonable combinatorics shows that, in $UL$, we have,</p> <p>$[C(x_1,\ldots, x_n), l] = \sum_{i=1}^n C(\ldots,[x_i,l],\ldots)$</p> <p>for $l\in L$. </p> <p>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. </p> http://mathoverflow.net/questions/54723/commutator-formulas-in-a-universal-enveloping-algebra/54777#54777 Answer by Stefan Waldmann for Commutator formulas in a universal enveloping algebra Stefan Waldmann 2011-02-08T15:34:23Z 2011-02-08T15:34:23Z <p>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).</p> <p>The symmetrization map can be viewed as a linear map $$\sigma\colon \mathrm{S}(L) \longrightarrow \mathrm{U}(L)$$ 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:</p> <p>What is the Gutt star product commutator of two (homogeneous) polynomials on $L^*$?</p> <p>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.</p> <p>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 <em>formal</em> 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 $$e_{\hbar x} \star_{\mathrm{Gutt}} e_{\hbar y} = e_{\mathrm{BHC}(\hbar x, \hbar y)}$$ 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.</p> <p>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</p>