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\begin{center}
  {\LARGE\bfseries Quantum \(\gl_N\): Nilpotent Parts and PBW Coordinates}\\[-1pt]
  {\small
  \href{https://www.math.toronto.edu/drorbn}{Dror Bar-Natan}:
  \href{https://drorbn.net/AcademicPensieve/}{AcademicPensieve}:
  \href{https://drorbn.net/AcademicPensieve/Projects/}{Projects}:
  \href{https://drorbn.net/AcademicPensieve/Projects/glneps/}{glneps}:
  \href{https://drorbn.net/AcademicPensieve/Projects/glneps/ChatGPT/}{ChatGPT}:
  \href{https://drorbn.net/AcademicPensieve/Projects/glneps/ChatGPT/qgln-260521-07.pdf}{qgln-260521-07.pdf}}
\end{center}
\vspace{-0.5em}
\hrule
\vspace{0.7em}

This is a convention sheet for the positive and negative nilpotent parts of the
Drinfeld--Jimbo quantum group.  The point is that, after choosing an order of roots,
these quantum nilpotent parts are \emph{visibly polynomial as vector spaces}.

\section*{1. Strictly upper-triangular generators}

Let \(\uq(\np)\) denote the positive nilpotent part.  We write its root vectors as
\[
  x_{ab}\qquad (1\leq a<b\leq N),
\]
where the simple generators are
\[
  x_{a,a+1}\qquad (1\leq a<N).
\]
For \(a<b\), define recursively
\begin{equation}\label{eq:xab-recursion}
  x_{ab}=x_{ac}x_{cb}-q^{-1}x_{cb}x_{ac}
       =\qcomm{x_{ac}}{x_{cb}},
  \qquad a<c<b.
\end{equation}
With the usual Drinfeld--Jimbo conventions, the right-hand side is independent of the
choice of intermediate index \(c\).  Equivalently, one may take the adjacent recursion
\[
  x_{ab}=x_{a,a+1}x_{a+1,b}-q^{-1}x_{a+1,b}x_{a,a+1}.
\]

For example,
\[
  x_{13}=x_{12}x_{23}-q^{-1}x_{23}x_{12},
\]
and
\[
  x_{14}=x_{12}x_{24}-q^{-1}x_{24}x_{12}
        =x_{13}x_{34}-q^{-1}x_{34}x_{13}.
\]

\section*{2. Strictly lower-triangular generators}

Similarly, let \(\uq(\nm)\) denote the negative nilpotent part.  We label its root
vectors by the same positive-root indices as the upper ones:
\[
  y_{ab}\qquad (1\leq a<b\leq N),
\]
so that \(y_{ab}\) is the adjoint/lower-triangular partner of \(x_{ab}\).  Thus the
simple generators are \(y_{a,a+1}\), meaning the usual lowering generators paired with
\(x_{a,a+1}\).  The analogous recursion is
\begin{equation}\label{eq:yab-recursion}
  y_{ab}=y_{cb}y_{ac}-q^{-1}y_{ac}y_{cb},
  \qquad a<c<b.
\end{equation}
This is what one gets by applying an anti-involution/adjoint with
\(x_{ab}^{*}=y_{ab}\) to \eqref{eq:xab-recursion}.  For example,
\[
  y_{13}=y_{23}y_{12}-q^{-1}y_{12}y_{23}.
\]
Some authors use the opposite normalization or replace \(q^{-1}\) by \(q\); this changes
the named root vectors but not the PBW/vector-space statement below.

\section*{3. Convex orderings of roots}

A total ordering \(<\) on the set \(\PhiP\) of positive roots is called \emph{convex} if
whenever
\[
  \alpha,\beta,\alpha+\beta\in\PhiP
  \quad\text{and}\quad
  \alpha<\beta,
\]
then
\begin{equation}\label{eq:convex-order}
  \alpha<\alpha+\beta<\beta.
\end{equation}

For type \(A_{N-1}\), identify the positive root \(\varepsilon_a-\varepsilon_b\) with the
pair \((a,b)\), where \(a<b\).  A standard convex ordering is
\[
  (1,2)<(1,3)<\cdots<(1,N)<(2,3)<\cdots<(2,N)<\cdots<(N-1,N).
\]
Another common convex ordering is the column order
\[
  (1,2)<(2,3)<(1,3)<(3,4)<(2,4)<(1,4)<\cdots.
\]
Different convex orders give different PBW bases, but all give the same vector-space
conclusion.

\section*{4. PBW bases}

Fix a convex ordering of the positive roots.  Then the ordered monomials
\begin{equation}\label{eq:pbw-plus}
  \PBWprod_{a<b} x_{ab}^{m_{ab}},
  \qquad m_{ab}\in\mathbb Z_{\geq 0},
\end{equation}
form a basis of \(\uq(\np)\).  Here the arrow means: multiply the factors in the chosen
convex order.

Likewise, the ordered monomials
\begin{equation}\label{eq:pbw-minus}
  \PBWprod_{a<b} y_{ab}^{n_{ab}},
  \qquad n_{ab}\in\mathbb Z_{\geq 0},
\end{equation}
form a basis of \(\uq(\nm)\), with the chosen convention for ordering.

\section*{5. Polynomial vector-space identification}

Therefore there are explicit vector-space isomorphisms
\[
  \QQ(q)[X_{ab}:a<b]\xrightarrow{\ \sim\ }\uq(\np),
  \qquad
  \prod_{a<b}^{\longrightarrow}X_{ab}^{m_{ab}}
  \longmapsto
  \prod_{a<b}^{\longrightarrow}x_{ab}^{m_{ab}},
\]
and
\[
  \QQ(q)[Y_{ab}:a<b]\xrightarrow{\ \sim\ }\uq(\nm),
  \qquad
  \prod_{a<b}^{\longrightarrow}Y_{ba}^{n_{ab}}
  \longmapsto
  \prod_{a<b}^{\longrightarrow}y_{ab}^{n_{ab}}.
\]

These are \textbf{not} algebra isomorphisms to commutative polynomial rings.  They are
PBW, hence vector-space, identifications.  The multiplication on the right is the
noncommutative \(q\)-deformed multiplication, governed by straightening relations.


\section*{6. Universal \(R\)-matrix in the same PBW coordinates}

With the same convex ordering and the same paired root vectors \(x_{ab}\leftrightarrow y_{ab}\),
the universal \(R\)-matrix has the ordered factorization
\begin{equation}\label{eq:R-factorization}
  \mathcal R=\Rcart\,\Rnil,
  \qquad
  \Rnil=
  \PBWprod_{a<b}
  \qexp{q^{-2}}\!\left((q-q^{-1})\,x_{ab}\otimes y_{ab}\right).
\end{equation}
Here
\[
  \qexp{t}(z):=\sum_{m\geq0}\frac{z^m}{[m]_t!},
  \qquad
  [m]_t!:=\prod_{j=1}^{m}\frac{1-t^j}{1-t}.
\]
Equivalently, the nilpotent factor expands in the PBW basis as
\begin{equation}\label{eq:R-pbw-expansion}
  \Rnil=
  \sum_{(m_{ab})}
  \left(\prod_{a<b}^{\longrightarrow}
    \frac{(q-q^{-1})^{m_{ab}}}{[m_{ab}]_{q^{-2}}!}\right)
  \left(\PBWprod_{a<b} x_{ab}^{m_{ab}}\right)
  \otimes
  \left(\PBWprod_{a<b} y_{ab}^{m_{ab}}\right),
\end{equation}
where the sum is over all functions \((a,b)\mapsto m_{ab}\in\mathbb Z_{\geq0}\).
The order of the factors in both tensor components is the chosen convex order; using the
opposite convention for the lower PBW order moves the reversal into the displayed formula.

For \(\gl_N\), the Cartan factor may be written, in the diagonal basis \(h_1,\ldots,h_N\), as
\begin{equation}\label{eq:R-cartan}
  \Rcart=q^{\sum_{i=1}^{N} h_i\otimes h_i},
\end{equation}
up to the usual completion and normalization conventions.  In the \(\slN\) normalization
one replaces this by the factor determined by the inverse Cartan matrix.

\section*{7. Tiny case: \(N=3\)}

For \(N=3\), the upper nilpotent part has root vectors
\[
  x_{12},\qquad x_{13}=x_{12}x_{23}-q^{-1}x_{23}x_{12},
  \qquad x_{23}.
\]
For the convex order \((1,2)<(1,3)<(2,3)\), the PBW basis is
\[
  x_{12}^{a}x_{13}^{b}x_{23}^{c},
  \qquad a,b,c\geq 0.
\]
Hence, as a vector space,
\[
  \uq(\np)\cong \QQ(q)[X_{12},X_{13},X_{23}].
\]

\vfill
\hrule
\smallskip
{\footnotesize Standard references: Lusztig, \emph{Introduction to Quantum Groups};
Jantzen, \emph{Lectures on Quantum Groups}; Chari--Pressley, \emph{A Guide to Quantum Groups}.}

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