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\begin{document} %\latintext
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\title{An Unexpected Cyclic Symmetry of $\slneps$}

\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{
  Mathematisch Instituut\\
  Universiteit Leiden\\
  Niels Bohrweg 1\\
  2333 CA Leiden\\
  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 introduce $\slneps$, a one-parameter family of Lie algebras that encodes an approximation of the semi-simple Lie algebra $sl_n$ by solvable algebras (this is useful elsewhere; see~\cite{Blah}). We find that $\slneps$ has an unanticipated order $n$ automorphism $\Psi$. Why is it there?
\end{abstract}

\maketitle

\setcounter{tocdepth}{3}
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%\eject

\draftcut\section{Statement}

We start with some conventions, then define our main stars the Lie algebras $\glneps$ and $\slneps$ in completely explicit terms, then exhibit the completely obvious automorphism $\Psi$ of $\glneps$ and of $\slneps$, and then go back to the abstract origins of $\glneps$ and $\slneps$, where the presence of $\Psi$ becomes surprising and unexplained.

\begin{convention} Let $n$ be a fixed positive integer and let $\eps$ be a formal parameter. For the formal symbol $x_{ij}$ where $1\leq i\neq j\leq n$ define its ``length'' $\lambda(x_{ij})\coloneqq\begin{cases} j-i & i<j \\ n-(i-j) & i>j \end{cases}$. Let $\chi_\eps$ be the function that assigns $\eps$ to \verb$True$ and $1$ to \verb$False$. For example, $\chi(5<7)=\eps$ while $\chi(7<5)=1$. Let $\delta_{ij}$ be the Kronecker $\delta$-function.
\end{convention}

\begin{definition} \label{def:glneps} Let $\glneps$ be the Lie algebra with generators $\{x_{ij}\}_{1\leq i\neq j\leq n} \cup \{a_i,b_i\}_{1\leq i\leq n}$ and with commutation relations
\begin{equation} \label{eq:cr}
  \begin{aligned}
    [x_{ij},x_{kl}] &= \chi_\eps(\lambda(x_{ij})+\lambda(x_{kl})>n)(\delta_{jk}x_{il}-\delta_{li}x_{kj})
      & \text{unless both }j=k\text{ and }l=i, \\
    [x_{ij},x_{ji}] &= \frac12(b_i-b_j+\eps(a_i-a_j)), & \\
    [a_i,x_{jk}] &= (\delta_{ij}-\delta_{ik})x_{jk}, & \\
    [b_i,x_{jk}] &= \eps(\delta_{ij}-\delta_{ik})x_{jk}. &
  \end{aligned}
\end{equation}
Let $\slneps$ be the subalgebra of $\glneps$ generated by the $x_{ij}$'s and by the differences $\{a_i-a_j\}$ and $\{b_i-b_j\}$. \endpar{\ref{def:glneps}}
\end{definition}

It is easy to verify that the commutation relations in~\eqref{eq:cr} respect the Jacobi identity and hence $\glneps$ and $\slneps$ are indeed Lie algebras.

It is easy to verify that if $\eps=1$ then $t_i\coloneqq b_i-a_i$ is central in $\glneps$, that $gl_{n+}^1\cong\langle x_{ij},h_i=(b_i+a_i)/2\rangle\oplus\langle t_i\rangle$, and that the first summand, $\langle x_{ij},h_i\rangle$, is isomorphic to the general linear Lie algebra $gl_n$ by mapping $x_{ij}$ to the matrix that has $1$ in position $(ij)$ and $0$ everywhere else and $h_i$ to the diagonal matrix that has $1$ in position $(ii)$ and $0$ everywhere else. Hence $\glneps$ is an $\eps$-dependent variant of $gl_n$ plus an Abelian summand, explaining its name $\glneps$. Nearly identical observations hold for $\slneps$: at $\eps=1$ it is a sum of $\sl_n$ and an Abelian summand.

It is also easy to verify that the map $\Phi_\eps\colon\glneps\to gl_{n+}^1$ defined by $a_i\mapsto a_i$, $b_i\mapsto\eps b_i$, and $x_{ij}\mapsto\chi_\eps(i>j)x_{ij}$ is a morphism of Lie algebras, and it is clearly invertible if $\eps$ is invertible. Hence for invertible $\eps$ our $\glneps$ is always a sum of $gl_n$ with an Abelian factor, and so $\glneps$ is most interesting at $\eps=0$ or in a formal neighborhood of $\eps=0$ (namely, over a ring like $\bbQ[\eps]/(\eps^{k+1}=0)$). Similarly for $\slneps$.

\begin{theorem} \label{thm:Psi} The map $\Psi\colon\glneps\to\glneps$ which increments all indices modulo $n$ is a Lie-algebra automorphism of $\glneps$ and/or $\slneps$. (Precisely, if $\psi$ is the single-cycle permutation $\psi=(123\ldots n)$ then $\Psi$ is defined by $\Psi(x_{ij})=x_{\psi(i)\psi(j)}$, $\Psi(a_i)=a_{\psi(i)}$, and $\Psi(b_i)=b_{\psi(i)}$).
\end{theorem}

\begin{proof} By case checking the length $\lambda(x_{ij})$ is $\Psi$-invariant, and hence everything in~\eqref{eq:cr} is $\Psi$-equivariant. \qed
\end{proof}

Thus our main theorem is a complete triviality. More precisely, Definition~\ref{def:glneps} was set up so that Theorem~\ref{thm:Psi} would be a complete triviality. But $\glneps$ also has an abstract origin which we describe in the next section. From the abstract perspective the existence of $\Psi$ remains mysterious to us.

Note that at $\eps=1$ the automorphism $\Psi$ induces an inner automorphism of $gl_n$ --- namely, it is conjugation $C(P_\psi)$ by a permutation matrix $P_\psi$. But at other values of $\eps$ the automorphism $\Psi$ does not restrict to conjugation by $P_\psi$, and at/near our point of interest $\eps=0$ the phrase ``conjugation by $P_\psi$'' stops making sense --- namely, $\lim_{\eps\to 0}\Phi_{1/\eps}\circ C(P_\psi)\circ\Phi_\eps$ does not exist.

Finally, for any permutation $\sigma$ one may define a map $A_\sigma\colon\glneps\to\glneps$ by permuting the indices: $A_\sigma(x_{ij})=x_{\sigma(i)\sigma(j)}$, $A_\sigma(a_i)=a_{\sigma(i)}$, and $A_\sigma(b_i)=b_{\sigma(i)}$). At $\eps=1$ the map $A_\sigma$ always respects the Lie bracket. Yet it is easy to verify that at $\eps\neq 1$ the only permutations $\sigma$ for which $A_\sigma$ is a morphism of Lie algebras are the powers of $\psi$, and obviously, $A_{\psi^p}=\Psi^p$.

\section{Wherefore $\glneps$?}

Let $\bbF$ be some ground field. A semi-simple Lie algebra over $\bbF$ (say, $gl_n$ or $sl_n$) can be reconstructed from its half: if $\frakg$ is a semi-simple Lie algebra (say, $gl_n$ or $sl_n$) and $\frakb^+$ is an upper Borel subalgebra (the upper triangular matrices if $\frakg$ is $gl_n$ or $sl_n$), then $\frakg$ can be recovered from $\frakb^+$, which has roughly half the dimension of $\frakg$.

Let us go through the process in some detail. The ``half'' $\frakb^+$ has its own Lie bracket $B^+\colon\frakb^+\otimes\frakb^+\to\frakb^+$. In addition, $\frakb^+$ is dual to a lower Borel subalgebra $\frakb^-$ (lower triangular matrices for $gl_n$ or $sl_n$, with the duality pairing $P\colon\frakb^-\otimes\frakb^+\to\bbF$ given by $P(L,U)=\tr(LU)$). Now $\frakb^-$ also has a bracket $B^-\colon\frakb^-\otimes\frakb^-\to\frakb^-$ and its adjoint relative to the duality $P$ is a ``cobracket'' map $\delta\colon\frakb^+\to\frakb^+\otimes\frakb^+$ which satisfies three conditions:
\begin{itemize}
\item $\delta$ is anti-symmetric: $\delta+\sigma\circ\delta=0$, where $\sigma\colon\frakb^+\otimes\frakb^+\to\frakb^+\otimes\frakb^+$ swaps the two tensor factors.
\item $\delta$ satisfies a co-Jacobi identity: $(1+\tau+\tau^2)\circ(1\otimes\delta)\circ\delta=0$, where $\tau\colon\frakb^+\otimes\frakb^+\otimes\frakb^+\to\frakb^+\otimes\frakb^+\otimes\frakb^+$ is the cyclic permutation of the tensor factors.
\item Along with the bracket $[\cdot,\cdot]=B^+$, $\delta$ satisfies a cocycle identity:
\[ \forall x_1,x_2\in\frakb^+,\quad
  \delta([x_1,x_2]) = (\ad_{x_1}\otimes 1+1\otimes\ad_{x_1})(\delta(x_2)) - (\ad_{x_2}\otimes 1+1\otimes\ad_{x_2})(\delta(x_1)).
\]
\end{itemize}

One may show that given any finite dimensional Lie algebra $\frakb$ and a co-bracket $\delta\colon\frakb\to\frakb\otimes\frakb$ satisfying the above conditions, then the adjoint $\delta^\ast\colon\frakb^\ast\otimes\frakb^\ast\to\frakb^\ast$ of the cobracket defines a bracket on $\frakb$, and then the ``double'' $\calD(\frakb,\delta)=\frakb\oplus\frakb^\ast$ is also a Lie algebra, with bracket
\[ [x_1\oplus y_1,x_2\oplus y_2] \coloneqq
  ([x_1,x_2]-\ad_{y_1}(x_2)+\ad_{y_2}(x_1)) \oplus ([y_1,y_2]+\ad_{x_1}(y_2)-\ad_{x_2}(y_1)),
\]
where we have used $\ad_x(y)$ to denote the coadjoint action of $\frakb$ on $\frakb^\ast$ (whose definition depends only on the bracket of $\frakb$) and $\ad_y(x)$ to denote the coadjoint action of $\frakb^\ast$ on $\frakb^{\ast\ast}=\frakb$ (whose definition depends only on the bracket of $\frakb^\ast$ or the cobracket of $\frakb$).

With all this in mind, $\frakg_+\coloneqq\calD(\frakb^+)\cong\frakg\oplus\frakh$, where $\frakh$ is an ``extra'' copy of the Cartan subalgebra of $\frakg$. In the case of $gl_n$ (and similarly for $sl_n$), this becomes the statement that the upper triangular matrices direct sum the lower triangular matrices make all matrices, but with the diagonal matrices repeated twice.

Clearly, if $\delta$ satisfies the three conditions above, then so does $\eps\delta$, where $\eps$ is any scalar. Hence we can define $\frakg_+^\eps\coloneqq\calD(\frakb^+,\eps\delta)$. At $\eps=1$ this is $\frakg\oplus\frakh$, and by scaling the $(\frakb^+)^\ast=\frakb^-$ component of $\calD(\frakb^+,\eps\delta)$ by a factor of $\eps$, the same is true whenever $\eps$ is invertible. Yet the one-parameter family $\frakg_+^\eps$ is not constant: at $\eps=0$ the double construction degenerates to the semi-direct product $\frakb^+\ltimes(\frakb^+)^\ast$, where $(\frakb^+)^\ast$ is taken as an Abelian Lie algebra and $\frakb^+$ acts on $(\frakb^+)^\ast$ using the coadjoint action. A Borel subalgebra is solvable, and its semi-direct product with an Abelian factor remains solvable. Hence $\frakg^0_+$ cannot be isomorphic to $\frakg\oplus\frakh$.

It is an elementary exercise to verify that if $\frakg=gl_n$ or $\frakg=sl_n$ then the resulting $\frakg_+^\eps$ is indeed $\glneps$ or $\slneps$ of the previous section.

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\end{document}
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