Let $\mathfrak g$ be a (semisimple) Lie algebra, $\mathfrak b\subset \mathfrak g$ a Borel and $\mathfrak n = [\mathfrak b,\mathfrak b]$. Then I am interested in solving certain division problems in $U(\mathfrak n)$ on a computer. More specifically, we recall $U(\mathfrak n)$ admits a weight decomposition into components $U(\mathfrak n)[\xi]$. Then given a $P\in U(\mathfrak n)[\xi]$ and $Q\in U(\mathfrak n)[\xi']$ for some $\xi,\xi'$, I wish to find a $P'\in U(\mathfrak n)[\xi'-\xi]$ solving $$ PP'=Q $$ The product here is the ordinary concatenation product. I know a (unique) solution exists from theoretical considerations. Furthermore all the coefficients are integers (in the PBW basis of $U(\mathfrak n)$, taking a Chevalley basis of $\mathfrak n$).
My current approach is sort of a brute force approach, and is unfortunately not very fast. I construct a (PBW) basis $\{f_i\}$ of $U(\mathfrak n)[\xi'-\xi]$, and a similar basis $\{g_j\}$ of $U(\mathfrak n)[\xi']$. Then for each $i$ we compute the product $$ A_i=Pf_i=\sum_jA_{ij} g_j $$ Giving a matrix $A$. At the same time we define a vector $b$ by $$ Q=\sum_jb_j g_j $$ Then we compute $P'$ by solving the linear problem $A^\top v=b$, and setting $P'=\sum_iv_if_i$.
I know in the literature there is some theory of non-commutative Gröbner bases, in particular also for universal enveloping algebras. I feel like such theory might help to solve these kind of problems, but I don't immediately see how.
