uNminus -- computes the universal enveloping algebra of the Lie algebra $N^{-}$

Let $\mathfrak{g}$ be a Lie algebra, let $\Phi^{+}$ be a set of positive roots, and let $N^{-}$ be the subalgebra spanned by the negative root vectors $\mathfrak{g}_{-\alpha}$. Let $T(N^{-})$ be the tensor algebra on $N^{-}$. The universal enveloping algebra $U(N^{-1})$ is the quotient of $T(N^{-})$ by the two-sided ideal generated by all relations of the form $Y_1 Y_2 - Y_2 Y_1 - [Y_1,Y_2]$.

We construct $T(N^{-})$ and $U(N^{-})$ using the AssociativeAlgebras package. The generators of $T(N^{-})$ are the negative root vectors in a Lie algebra basis of $\mathfrak{g}$. Let $Y_1,\ldots,Y_n$ the negative root vectors. Then the term order on $T(N^{-})$ we use is degree lex with the variables ordered $Y_l,\ldots,Y_1$, so that the basis used for the quotient $U(N^{-})$ consists of monomials of the form $Y_1^{a_1}\cdots Y_l^{a_l}$.

In the following example, we express the monomial $Y_3 Y_2 Y_1$ in $U(N^{-})$ for $\mathfrak{g} = sl_3$. We have $Y_1 = E_{(2,1)}$, $Y_2 = E_{(3,2)}$ and $Y_3 = E_{(3,1)}$. Thus $Y_2 Y_1 = Y_1 Y_2+Y_3$ and $Y_3 Y_1 = Y_1 Y_3$, so that $Y_3 Y_2 Y_1 = Y_1 Y_2 Y_3+Y_3^2$.