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\def\lightred#1{\setlength{\fboxsep}{0pt}\colorbox{lightred}{#1}} \def\gray{\color{gray}} \def\red{\color{red}} \def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}} \def\yellowt#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{#1}}} % Following % http://tex.stackexchange.com/questions/23521/tabular-vertical-alignment-to-top \def\imagetop#1{\vtop{\null\hbox{#1}}} \def\ds{\displaystyle} \def\eps{\epsilon} \def\slneps{{sl_{n+}^\eps}} \def\glneps{{gl_{n+}^\eps}} \def\endpar#1{~\hfill\fbox{\footnotesize#1}} \begin{document} %\latintext \newdimen\captionwidth\captionwidth=\hsize \setcounter{secnumdepth}{4} \title{An Unexpected Cyclic Symmetry of $I\fraku_n$} \author{Dror~Bar-Natan} \address{ Department of Mathematics\\ University of Toronto\\ Toronto Ontario M5S 2E4\\ Canada } \email{drorbn@math.toronto.edu} \urladdr{http://www.math.toronto.edu/drorbn} \author{Roland~van~der~Veen} \address{ University of Groningen, Bernoulli Institute\\ P.O. Box 407\\ 9700 AK Groningen\\ The Netherlands } \email{roland.mathematics@gmail.com} \urladdr{http://www.rolandvdv.nl/} \date{First edition Not Yet, 2020, this edition \today.} \subjclass[2010]{57M25} \keywords{ Lie algebras, Lie bi-algberas, solvable approximation } \thanks{This work was partially supported by NSERC grant RGPIN-2018-04350.} \begin{abstract} We find, discuss, and extend an unexpected (to us) order $n$ cyclic group of automorphisms of the Lie algebra $I\fraku_n\coloneqq\fraku_n\ltimes\fraku_n^\ast$, where $\fraku_n$ is the Lie algebra of upper triangular $n\times n$ matrices. \end{abstract} \maketitle %\setcounter{tocdepth}{3} %\tableofcontents %\eject Given any Lie algebra $\fraka$ one may form its ``inhomogeneous version'' $I\fraka\coloneqq\fraka\ltimes\fraka^\ast$, its semidirect product with its dual $\fraka^\ast$ where $\fraka^\ast$ is considered as an Abelian Lie\footnote{Two Norwegians!} algebra and $\fraka$ acts on $\fraka^\ast$ via the coadjoint action. (Over $\bbR$ if $\fraka=so_3$ then $\fraka^\ast=\bbR^3$ and so $I\fraka=so_3\ltimes\bbR^3$ is the Lie algebra of the Euclidean group of rotations and translations, explaining the name). In general, we care about $I\fraka$. It is the Drinfel'd double / Manin triple construction~\cite{Drinfeld:QuantumGroups, EtingofSchiffman:QuantumGroups} when the cobracket is $0$. These Lie algebras occur in the study of the Kashiwara-Vergne problem~\cite{Talk:Bonn, WKO2} and they provide the simplest quantum algebra context for the Alexander polynomial~\cite{Talk:Chicago, WKO1}. We care especially for the case where $\fraka$ is a Borel subalgebra of a semi-simple Lie algebra (e.g., upper triangular matrices) as then the algebras $I\fraka$ are the $\eps=0$ ``base case'' for ``solvable approximation''~\cite{PP1, DoPeGDO, DPG, Talk:SolvApp, Talk:Dogma, Talk:DoPeGDO}, and their automorphisms are expected to become symmetries of the resulting knot invariants. Let $\fraku_n$ be the Lie algebra of upper triangular $n\times n$ matrices. Beyond inner automorphisms, $\fraku_n$ and hence $I\fraku_n$ has one obvious and expected automorphism $\Phi$ corresponding to flipping matrices along their anti-main-diagonal, as shown in the first image of Figure~\ref{fig:ExpectedAndUnexpected}. With $x_{ij}$ denoting the $n\times n$ matrix with $1$ in position $(ij)$ and zero everywhere else ($i\leq j$ in $\fraku_n$), $\Phi$ is given by $x_{ij}\mapsto x_{n+1-j,n+1-i}$. \begin{figure} \[ \resizebox{!}{3cm}{\input{ExpectedAndUnexpected.pdf_t}} \] \caption{An expected automorphism (left), an unexpected one (middle), and an alternative presentation of the ``layers'' table (right).} \label{fig:ExpectedAndUnexpected} \end{figure} There clearly isn't an automorphism of $\fraku_n$ that acts by ``sliding down and right parallel to the main diagonal'', as in the second image in Figure~\ref{fig:ExpectedAndUnexpected}. Where would the last column go? Yet the sliding map, when restricted to where it is clearly defined ($\fraku_n$ with the last column excluded), does extend to an automorphism of $I\fraku_n$ as in the theorem below. While the proof is trivial, we don't feel that we fully understand Theorem~\ref{thm:Psi4Iu}. \begin{theorem} \label{thm:Psi4Iu} With the basis $\{x_{ij}\}_{1\leq ij \end{cases}$. It is easy to verify that the length $\lambda(x_{ij})$ is $\Psi$-invariant, and hence everything in~\eqref{eq:cr0} is $\Psi$-equivariant. \qed $I\fraku_n$ is a solvable Lie algebra (as a semi-direct product of solvable with Abelian, and as will be obvious from the table below). It is therefore interesting to look at the structure of its commutator subgroups. This structure is summarized in the following table (an alternative view is in Figure~\ref{fig:ExpectedAndUnexpected}): \needspace{45mm} \[ \arraycolsep=2pt \def\arraystretch{1.2} \begin{array}{|l|c|cccccccccc|} \hline \text{layer }0 & \frakg=I\fraku_n & a_1 & \to & a_2 & \to \cdots \to & a_{n-2} & \to & a_{n-1} & \to & a_n & \to \\ \text{layer }1 & \frakg'_1=\frakg'=[\frakg,\frakg] & x_{12} & \to & x_{23} & \to \cdots \to & x_{n-2,n-1} & \to & x_{n-1,n} & \to & x_{n1} & \to \\ \text{layer }2 & \frakg'_2=[\frakg',\frakg'_1] & x_{13} & \to & x_{24} & \to \cdots \to & x_{n-2,n} & \to & x_{n-1,1} & \to & x_{n2} & \to \\ \text{layer }3 & \frakg'_3=[\frakg',\frakg'_2] & x_{14} & \to & x_{25} & \to \cdots \to & x_{n-2,1} & \to & x_{n-1,2} & \to & x_{n3} & \to \\ \vdots & \vdots & \vdots & & \vdots & & \vdots & & \vdots & & \vdots & \\ \text{layer }(n-1) & \frakg'_{n-1}=[\frakg',\frakg'_{n-2}] & x_{1n} & \to & x_{21} & \to \cdots \to & x_{n-2,n-1} & \to & x_{n-1,n-2} & \to & x_{n,n-1} & \to \\ \text{layer }n & \frakg'_n=[\frakg',\frakg'_{n-1}] & b_1 & \to & b_2 & \to \cdots \to & b_{n-2} & \to & b_{n-1} & \to & b_n & \to \\ \hline \end{array} \] In this table (all assertions are easy to verify): \begin{itemize}[leftmargin=*,labelindent=0pt] \item Apart for the treatment of the $a_i$'s and the $b_i$'s, layer$=$length. \item The layers indicate a filtration; each layer should be considered to contains all the ones below it. The generators marked at each layer generate it modulo the layers below. \item The bracket of an element at layer $p$ with an element of layer $q$ is in layer $p+q$ (and it must vanish if $p+q>n$). \item If $p\geq 2$, every generator in layer $p$ is the bracket of a generator in layer $1$ with a generator in layer $p-1$. \item In layer $p$, the first $n-p$ generators indicated belong to $\fraku_n$ and the last $p$ belong to $\fraku_n^\ast$. So as we go down, $\fraku_n^\ast$ slowly ``overtakes'' the table. \item The automorphism $\Psi$ acts by following the arrows and shifting every generator one step to the right (and pushing the rightmost generator in each layer back to the left). \item The automorphism $\Phi$ acts by mirroring the $\fraku_n$ part of each layer left to right and by doing the same to the $\fraku_n^\ast$ part, without mixing the two parts. \item Note that $I\fraku_n$ can be metrized by pairing the $\fraku_n$ summand with the $\fraku_n^\ast$ one. The metric only pairs generators indicated in layer $p$ with generators indicated in layer $(n-p)$. \end{itemize} Note also that the brackets of the generators indicated in layer 1 yield the generators indicated in layer 2 as follows: \[ \xymatrix@R=10pt@C=10pt{ x_{12} \ar[rd] && x_{23} \ar[ld]\ar[rd] && x_{34} \ar[ld]\ar[rd] && \cdots && x_{n-1,n} \ar[ld]\ar[rd] && x_{n1} \ar[ld]\ar[rd] && \ar[ld] \\ & x_{13} && x_{24} &&& \cdots &&& x_{n-1,1} && x_{n2} & } \] (with the diagram continued cyclically). The symmetry group of the above cycle is the dihedral group $D_n$ and this strongly suggests that the group of outer automorphisms of $I\fraku_n$ (all automorphisms modulo inner ones) is $D_n$, generated by $\Phi$ and $\Psi$. We did not endeavor to prove this formally. \noindent{\bf Extension.} As already mention, $I\fraku_n$ is the result of the Drinfel'd double / Manin triple construction~\cite{Drinfeld:QuantumGroups, EtingofSchiffman:QuantumGroups} on $\fraku_n$ with vanishing cobracket. But $\fraku_n$ has another cobracket $\delta$ which it gains as the dual of the Lie algebra $\frakl_n$ of lower triangular matrices. With $\eps$ a formal parameter, one may form the double of $\fraku_n$ using the cobracket $\eps\delta$. The result is a Lie algebra $gl_{n+}^\eps$ over the ring of polynomials is $\eps$ which specializes to $I\fraku_n$ at $\eps=0$ and which is isomorphic to $gl_n\oplus\frakh'_n$ when $\eps$ is invertible, where $\frakh'_n$ denotes a second copy of the diagonal matrices in $gl_n$. We care about $gl_{n+}^\eps$ a lot~\cite{PP1, DoPeGDO, DPG, Talk:SolvApp, Talk:Dogma, Talk:DoPeGDO}. \begin{theorem} With the same conventions as in Theorem~\ref{thm:Psi4Iu} the map $\Psi$ is also a Lie algebra automorphism of $gl_{n+}^\eps$. \end{theorem} \begin{proof} By easy case checking and explicit computations, the commutation relations of $gl_{n+}^\eps$ are given by \[ \begin{aligned} [x_{ij},x_{kl}] &= \chi_\eps(\lambda(x_{ij})+\lambda(x_{kl})