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\def\web#1{\href{\weburl/#1}{\begin{greek}web\end{greek}/#1}} \def\bbe{{\mathbb e}} \def\bbA{{\mathbb A}} \def\bbB{{\mathbb B}} \def\bbC{{\mathbb C}} \def\bbD{{\mathbb D}} \def\bbE{{\mathbb E}} \def\bbF{{\mathbb F}} \def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbV{{\mathbb V}} \def\calB{{\mathcal B}} \def\calD{{\mathcal D}} \def\calF{{\mathcal F}} \def\calG{{\mathcal G}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}} \def\calR{{\mathcal R}} \def\calU{{\mathcal U}} \def\fraka{{\mathfrak a}} \def\frakb{{\mathfrak b}} \def\frakd{{\mathfrak d}} \def\frakg{{\mathfrak g}} \def\frakh{{\mathfrak h}} \def\frakl{{\mathfrak l}} \def\frakn{{\mathfrak n}} \def\fraku{{\mathfrak u}} \def\hatcalU{\hat{\mathcal U}} \def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}} \def\Hom{\operatorname{Hom}} \def\Id{\text{\it Id}} \def\draftcut{\if\draft y \cleardoublepage \fi} %\def\draftcut{\cleardoublepage} \definecolor{lightred}{RGB}{255, 217, 217} \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 February 3, 2020, this edition \today.} \subjclass[2010]{57M25} \keywords{ Lie algebras, Lie bialgebras, Lie algebra automorphism, solvable approximation, triangular matrices} \thanks{This work was partially supported by NSERC grant RGPIN-2018-04350. It is available in electronic form, along with source files and a verification {\sl Mathematica} notebook at \url{http://drorbn.net/UnexpectedCyclic} and at \arXiv{2002.00697}.} \begin{abstract} We find and discuss 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. Our results also extend to $gl_{n+}^\eps$, a ``solvable approximation'' of $gl_n$, as defined within. \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 a special case of 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 anti-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 anti-automorphism (left), an unexpected automorphism (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$=\lambda(x_{ij})$. \item The layers indicate a filtration; each layer should be considered to contain 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 anti-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 and anti-automorphisms of $I\fraku_n$ (all automorphisms and anti-automorphisms modulo inner automorphisms) is $D_n$, generated by $\Phi$ and $\Psi$. We did not endeavor to prove this formally. \vskip 2mm\noindent{\bf Extension.} The Drinfel'd double / Manin triple construction~\cite{Drinfeld:QuantumGroups, EtingofSchiffman:QuantumGroups}, when applied to $\fraku_n$, is a way to reconstruct $gl_n$ from its subalgebras of upper triangular matrices $\fraku_n$ and lower triangular matrices $\frakl_n$. Precisely, one endows the vector space $\frakg=\fraku_n\oplus\frakl_n$ with a non-degenerate symmetric bilinear form by declaring that the subspaces $\fraku_n$ and $\frakl_n$ are isotropic ($\langle\fraku_n,\fraku_n\rangle=\langle\frakl_n,\frakl_n\rangle=0$) and by setting $\langle x_{kl},x_{ij}\rangle=\delta_{li}\delta_{jk}$, $\langle b_i,a_j\rangle=2\delta_{ij}$, and $\langle x_{ji},a_k\rangle = \langle b_k,x_{ij}\rangle = 0$ as in Theorem~\ref{thm:Psi4Iu} and where $a_i$ stands for the diagonal matrix $x_{ii}$ considered as an element of $\fraku_n$ and $b_i$ stands for the same matrix as an element of $\frakl_n$. There is then a unique bracket on $\frakg$ that extends the brackets on the summands $\fraku_n$ and $\frakl_n$ and relative to which the inner product of $\frakg$ is invariant. With our judicious choice of bilinear form, this bracket on $\frakg$ satisfies the Jacobi identity and turns $\frakg$ into a Lie algebra isomorphic to $gl_{n+}=gl_n\oplus\frakh'_n$, where $\frakh'_n$ denotes a second copy of the diagonal matrices in $gl_n$. We let $gl_{n+}^\eps$ be the Inonu-Wigner~\cite{InonuWigner:ContractionOfGroups} contraction of $\frakg$ along its $\frakl_n$ summand, with parameter $\epsilon$.\footnote{Alternatively, make $\fraku_n$ into a Lie bialgebra with cobracket $\delta$ using its given duality with $\frakl_n$, and double it as in~\cite{Drinfeld:QuantumGroups, EtingofSchiffman:QuantumGroups} but using the cobracket $\eps\delta$.} All that this means is that the bracket of $\frakl_n$ gets multiplied by $\epsilon$ to give $\frakl_n^\epsilon$, and then the Drinfel'd double / Manin triple construction is repeated starting with $\fraku_n\oplus\frakl_n^\eps$, without changing the bilinear form. The result is a Lie algebra $gl_{n+}^\eps$ over the ring of polynomials in $\eps$ which specializes to $I\fraku_n$ at $\eps=0$ and which is isomorphic to $gl_n\oplus\frakh'_n$ when $\eps$ is invertible.\footnote{Hence $gl_{n+}^\eps\to I\fraku_n$ is a counter-example to the feel-true statement ``a contraction of a direct sum is a direct sum''. Indeed with notation as in Theorem~\ref{thm:Psi4gleps}, as $\eps\to 0$ the decomposition $gl_{n+}^\eps = gl_n\oplus\frakh'_n = \langle x_{ij},b_i+\eps a_i\rangle \oplus \langle b_i-\eps a_i\rangle$ collapses.}\ We care about $gl_{n+}^\eps$ a lot~\cite{PP1, DoPeGDO, DPG, Talk:SolvApp, Talk:Dogma, Talk:DoPeGDO}; when reduced modulo $\eps^{k+1}=0$ for some natural number $k$ it becomes solvable, and hence a ``solvable approximation'' of $gl_n$ with applications to computability of knot invariants. \begin{theorem} \label{thm:Psi4gleps} 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 some 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})