Infinitesimal Modular Group: $q$-Deformed $\mathfrak{sl}_{2}$ and Witt Algebra

Infinitesimal Modular Group:
$\boldsymbol{q}$-Deformed $\boldsymbol{\mathfrak{sl}_{2}}$ and Witt Algebra

Alexander THOMAS

A. Thomas

Universität Heidelberg, Berliner Str. 41-49, 69120 Heidelberg, Germany
<EMAIL_ADDRESS>https://thomas-math.wixsite.com/maths/en

Received December 01, 2023, in final form June 03, 2024; Published online June
20, 2024

We describe new $q$-deformations of the 3-dimensional Heisenberg algebra, the
simple Lie algebra $\mathfrak{sl}_{2}$ and the Witt algebra. They are
constructed through a realization as differential operators. These operators
are related to the modular group and $q$-deformed rational numbers defined by
Morier-Genoud and Ovsienko and lead to $q$-deformed Möbius transformations
acting on the hyperbolic plane.

quantum algebra; Lie algebra deformations; $q$-Virasoro; Burau representation

35A01; 65L10; 65L12; 65L20; 65L70

##  1 Introduction and results

The construction of $q$-deformed rational numbers by Morier-Genoud and
Ovsienko [12] starts from the observation that rational numbers are generated
by the image of zero under the action of the modular group
$\operatorname{PSL}_{2}(\mathbb{Z})$. This group is generated by the
translation $T(x)=x+1$ and the inversion $S(x)=-1/x$. The only relations
between these operations are $S^{2}=\mathrm{id}=(ST)^{3}$.

The $q$-deformed integers
$[n]_{q}=1+q+q^{2}+\dots+q^{n-1}=\frac{1-q^{n}}{1-q}$, where
$q\in\mathbb{C}^{*}$, satisfy the relation $[n+1]_{q}=q[n]_{q}+1$. It is
natural to introduce as $q$-analog to the translation $T$ the transformation
$T_{q}(x)=qx+1$. The map $S_{q}(x)=-1/(qx)$ satisfies
$S_{q}^{2}=\mathrm{id}=(S_{q}T_{q})^{3}$. The  _$q$ -rational numbers_ are
then defined by the image of zero under the action by $T_{q}$ and $S_{q}$
using for example the continued fraction representation of a rational number.
Since these operations are Möbius transformations, we can represent them in
matrix form as follows:

$T_{q}=\begin{pmatrix}q&1\\\ 0&1\end{pmatrix}\qquad\text{and}\qquad
S_{q}=\begin{pmatrix}0&-1\\\ q&0\end{pmatrix}.$

This coincides with the _reduced Burau representation_ of the braid group
$B_{3}$ with parameter ${t=-q}$ [4]. Indeed the standard generators of $B_{3}$
are represented by $\sigma_{1}=T_{q}$ and
$\sigma_{2}=S_{q}T_{q}S_{q}=\left(\begin{smallmatrix}1&0\\\
-q&q\end{smallmatrix}\right)$. One easily checks the braid relation
$\sigma_{1}\sigma_{2}\sigma_{1}=\sigma_{2}\sigma_{1}\sigma_{2}$. The
faithfulness of specializations of the Burau representation (where $q$ is not
a formal parameter, but a non-zero complex number) is an open question [3,
Section 7]. It was studied for real values of $q$ in [18]. In [14], a link to
$q$-deformed rational numbers allows to partially solve the open question.

Using Taylor expansions of $q$-rational numbers, one can define $q$-real
numbers [13] which are power series in $q$ with integer coefficients. A
natural question is _how to do analysis with these $q$-real numbers?_ Basic
functions on real numbers are monomials and the exponential function, which
are eigenfunctions of the vector fields associated to
$\mathfrak{sl}_{2}(\mathbb{R})$ (acting on the completed line
$\mathbb{RP}^{1}$). The goal of our investigation is to $q$-deform these
vector fields and to analyze their eigenfunctions.

Following a suggestion of Valentin Ovsienko, we can associate to $T_{q}$ a
differential operator $D_{-1}(q)$, which corresponds to the infinitesimal
$q$-shift. For $q=1$, we have ${D_{-1}(1)=\partial={\rm d}/{\rm d}x}$. This
operator is given by

$D_{-1}:=(1+(q-1)x)\partial.$

One can directly check that $D_{-1}$ commutes with $T_{q}$, where $T_{q}$ acts
on the space of functions by precomposition. The starting point of the paper
is the question whether there is a differential operator associated to
$S_{q}$. This would allow to define in some sense a Lie algebra for the
modular group $\operatorname{PSL}_{2}(\mathbb{Z})$, or an infinitesimal
version of the Burau representation of $B_{3}$.

In the classical setting for $q=1$, there is an operator which _anti-commutes_
with $S$:

$S\circ x\partial+x\partial\circ S=0,$

where $S$ acts on the space of functions by precomposition. We introduce the
differential operator $D_{0}$, a $q$-deformed version of $x\partial$, given by

$D_{0}:=(1+(x-1)q)D_{-1}=(1+(x-1)q)(1+(q-1)x)\partial.$

We will see that $D_{0}$ anti-commutes with $S_{q}$. Together with
$D_{1}:=S_{q}\circ D_{-1}\circ S_{q}$, a deformation of $x^{2}\partial$, we
get three differential operators which are closed under the bracket (see
Theorem 2.3):

######  Theorem 1.1.

The operators $D_{-1}$, $D_{0}$ and $D_{1}$ form a Lie algebra with brackets

$\displaystyle[D_{0},D_{1}]=\big{(}q^{2}-q+1\big{)}D_{1}+(1-q)D_{0},\qquad[D_{0},D_{-1}]=-\big{(}q^{2}-q+1\big{)}D_{-1}+(1-q)D_{0},$
$\displaystyle[D_{-1},D_{1}]=2D_{0}+(1-q)(D_{1}-D_{-1}).$

The theorem tells us that the module over $\mathbb{R}[q]$ generated by
$D_{-1}$, $D_{0}$ and $D_{1}$ is a deformation of the Lie algebra
$\mathfrak{sl}_{2}(\mathbb{R})$ which we recover for $q=1$. The Lie algebra
$\mathfrak{sl}_{2}$ being simple, it does not allow for non-trivial
deformations. Hence our deformation is isomorphic to $\mathfrak{sl}_{2}$ as a
Lie algebra, but they are different as $\mathbb{Z}[q]$-modules. This is
similar to quantum groups.

A fundamental role is played by the Möbius transformation

$g_{q}(x)=\frac{1+(x-1)q}{1+(q-1)x},$

which is a deformation of the identity. It is the eigenfunction of $D_{0}$
with eigenvalue $q^{2}-q+1$ and normalization $g_{q}(0)=1-q$. We call it the
_$q$ -rational transition map_ since it makes a passage between two different
$q$-deformations of rational numbers studied in [2]. More precisely (see
Theorem 2.7):

######  Theorem 1.2.

The two $q$-deformations of rational numbers defined in [2, Definition 2.6]
are linked via

$g_{q}\left(\left[\frac{r}{s}\right]_{q}^{\sharp}\right)=\left[\frac{r}{s}\right]_{q^{-1}}^{\flat}.$

This theorem comes from the interplay between $g_{q},T_{q}$ and $S_{q}$ given
by $g_{q}\circ T_{q}=T_{q^{-1}}\circ g_{q}$ and $g_{q}\circ
S_{q}=S_{q^{-1}}\circ g_{q}$ (see Proposition 2.6). The $q$-rational
transition map also satisfies a sort of duality between $q$ and $x$:

$g_{q}(x)g_{x}(q)=1.$

The map $g_{q}$, as well as its multiplicative inverse $g_{q}^{-1}=1/g_{q}$,
behave very well with the three operators $D_{-1}$, $D_{0}$ and $D_{1}$ (see
Propositions 2.11 and 3.1):

######  Proposition 1.3.

The $q$-rational transition map $g_{q}$ and the differential operators
$D_{-1}$, $D_{0}$, and $D_{1}$ interact in the following way:

  1. $(1)$

$D_{0}(g_{q})=\big{(}q^{2}-q+1\big{)}g_{q}$, $D_{-1}(g_{q})=q+(1-q)g_{q}$,
$D_{1}(g_{q})=(q-1)g_{q}+g_{q}^{2}$,

  2. $(2)$

$g_{q}D_{0}=(1-q)D_{0}+\big{(}q^{2}-q+1\big{)}D_{1}$,
$g_{q}D_{-1}=D_{0}+(1-q)D_{1}$,

  3. $(3)$

$qg_{q}^{-1}D_{0}=(q-1)D_{0}+\big{(}q^{2}-q+1\big{)}D_{-1}$,
$qg_{q}^{-1}D_{1}=D_{0}+(q-1)D_{-1}$.

These relations allow a deformation of the Witt algebra, the complexification
of the Lie algebra of polynomial vector fields on the circle. The Witt algebra
is described by a vector space basis $(\ell_{n})_{n\in\mathbb{Z}}$ with
bracket given by

$[\ell_{n},\ell_{m}]=(m-n)\ell_{n+m}.$

This algebra can be realized as differential operators (or equivalently as
vector fields) via ${\ell_{n}=x^{n-1}\partial}$. Putting for $n>1$:

$D_{n}=g_{q}^{n-1}D_{1}\qquad\text{and}\qquad
D_{-n}=\big{(}qg_{q}^{-1}\big{)}^{n-1}D_{-1},$

we get a deformation of the Witt algebra (see Theorem 3.2):

######  Theorem 1.4.

The $(D_{n})_{n\in\mathbb{Z}}$ form a Lie algebra with bracket given by
$($where $n,r>0)$:

$\displaystyle[D_{0},D_{n}]=n\big{(}q^{2}-q+1\big{)}D_{n}+\big{(}q^{2}-q+1\big{)}\displaystyle\sum_{k=1}^{n-1}(1-q)^{k}D_{n-k}+(1-q)^{n}D_{0},$
$\displaystyle[D_{n},D_{n+r}]=rD_{2n+r}+(q-1)rD_{2n+r-1},$
$\displaystyle[D_{-n},D_{n}]=2nq^{n-1}D_{0}+(2n-1)q^{n-1}(q-1)(D_{-1}-D_{1}),$
$\displaystyle[D_{n+r},D_{-n}]=(q-1)q^{n-1}(2n+r-1)D_{r+1}-\big{(}q^{2}+(2n+r-2)q+1\big{)}q^{n-1}D_{r},$
$\displaystyle\phantom{[D_{n+r},D_{-n}]=}{}-q^{n-1}\big{(}q^{2}-q+1\big{)}\displaystyle\sum_{k=1}^{r-1}(1-q)^{k}D_{r-k}-(1-q)^{r}q^{n-1}D_{0}.$

The remaining brackets $[D_{0},D_{-n}]$, $[D_{-n},D_{-n-r}]$ and
$[D_{n},D_{-n-r}]$ obey similar formulas.

Integrating the vector fields associated to $D_{-1}$, $D_{0}$ and $D_{1}$ on
the hyperbolic plane, we get Möbius transformations. We speculate about a
$q$-deformed hyperbolic plane on which these transformations naturally act.
The boundary of this deformed hyperbolic plane should be the $q$-deformed real
numbers. Other interesting open questions include the link between our
$q$-deformed $\mathfrak{sl}_{2}$ and the quantum group
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$, or the existence of a central extension
of our deformed Witt algebra, which would give a deformed Virasoro algebra.

Deformations of rational numbers were introduced in [12], extended to real
numbers in [13] and to Gaussian integers in [17]. Many different deformations
of the Witt algebra or its central extension, the Virasoro algebra, have been
introduced in the past: first in [6] and then in [5] deforming the matrix Lie
bracket to $[A,B]_{q}=qAB-q^{-1}BA$. This also deforms the Jacobi identity. A
similar construction was done in [9] viewing the Witt algebra as space of
derivations of $\mathbb{C}\big{[}x^{\pm 1}\big{]}$ and using the
$q$-differential $\partial_{q}(f)=\tfrac{f(qx)-f(x)}{qx-x}$. This was
generalized in [8] to more general $\sigma$-derivatives. Deforming the cocycle
gives a $q$-Virasoro algebra in [10], developed into a theory of $q$-deformed
pseudo-differential operators in [11]. A deformation as Lie algebra in terms
of an operator product expansion is given in [19]. A similar proposal can be
found in [7, equation (1.3)], using a $q$-deformed Miura transformation. In
[15, equation (38)], the deformation

$[T_{m}(q),T_{n}(q)]=([-n]_{q}-[-m]_{q})(T_{n+m}\big{(}q^{2}\big{)}-T_{n+m}(q))$

is studied. Yet another proposal from [16, formula (3.18)] gives operators
$D_{n}(q)$ for $n\in\mathbb{Z}$ with commutator
${[D_{n}(q),D_{m}(q)]=\big{(}q-q^{-1}\big{)}[n-m]_{q}D_{n+m}\big{(}q^{2}\big{)}}$
(removing the central extension). Finally, in [1], a two-dimensional
deformation using elliptic algebras is studied. All these approaches are
different from ours.

Structure of the paper. In Section 2, we introduce and study the deformation
of $\mathfrak{sl}_{2}$, the Heisenberg algebra and the $q$-rational transition
map. This is broadened in Section 3 to a deformed Witt algebra. In the final
Section 4, we study the Möbius transformations associated to these
deformations.

##  2 Deformed $\boldsymbol{\mathfrak{sl}_{2}}$ and Heisenberg algebra

The group $\mathrm{SL}_{2}$ acts naturally on the projective line
$\mathbb{P}^{1}$. We will work over $\mathbb{R}$ or $\mathbb{C}$.
Differentiating this action at the identity gives a realization of the Lie
algebra $\mathfrak{sl}_{2}$ as vector fields on $\mathbb{P}^{1}$. Using the
two standard charts of $\mathbb{P}^{1}$ with transition function $x\mapsto
1/x$, the image of
$\mathfrak{sl}_{2}\to\mathrm{Vect}\big{(}\mathbb{P}^{1}\big{)}$ is generated
by $\partial$, $x\partial$ and $x^{2}\partial$ written in the first chart,
where we use the notation $\partial={\rm d}/{\rm d}x$. One readily checks that
these expressions are well-defined over the second chart.

We construct a deformation of these three differential operators. They come as
a realization of a Lie algebra which itself deforms $\mathfrak{sl}_{2}$.
Together with a $q$-deformed identity map, we deform the 3-dimensional
Heisenberg algebra.

###  2.1 Deformed $\boldsymbol{\mathfrak{sl}_{2}}$

On $\mathbb{P}^{1}$, consider the Möbius transformations

$T_{q}(x)=qx+1\qquad\text{and}\qquad S_{q}(x)=-\frac{1}{qx},$

where $q\in\mathbb{C}^{*}$ is fixed or seen as a formal parameter. They deform
the translation $x\mapsto x+1$ and the inversion $x\mapsto-1/x$. These
transformations act on the space of functions on $\mathbb{P}^{1}$ by
precomposition.

Consider the differential operator $D_{-1}$ on $\mathbb{P}^{1}$ which is
defined in the first chart by

$D_{-1}:=(1+(q-1)x)\partial.$

######  Proposition 2.1.

The operators $D_{-1}$ and $T_{q}$ commute, where $T_{q}$ acts on the space of
functions by precomposition.

###### Proof.

For a function $f(x)$, we have on the one side

$D_{-1}\circ T_{q}(f(x))=D_{-1}(f(qx+1))=(1+(q-1)x)qf^{\prime}(qx+1).$

On the other side,

$T_{q}\circ
D_{-1}(f(x))=T_{q}((1+(q-1)x)f^{\prime}(x))=(1+(q-1)(qx+1))f^{\prime}(qx+1).$

Both expressions coincide. ∎

The unique eigenfunction $E_{q}$ of $D_{-1}$ with eigenvalue 1 and
normalization $E_{q}(0)=1$ is a $q$-deformation of the exponential function,
called the _Tsallis exponential_ [20]. This was first observed by Valentin
Ovsienko and Emmanuel Pedon.111Unpublished, private communication. To find
$E_{q}$, one has to solve ${f=D_{-1}f=(1+(q-1)x)f^{\prime}}$, i.e., $(\ln
f)^{\prime}=\frac{1}{1+(q-1)x}$. The solution is given by

$E_{q}(x)=(1+(q-1)x)^{\frac{1}{q-1}}.$

It satisfies $E_{q}(qx+1)=E_{q}(1)E_{q}(x)$ since $E_{q}(qx+1)=T_{q}E_{q}$ is
also an eigenfunction of $D_{-1}$ with eigenvalue 1.

The main new operator we introduce is the following:

$D_{0}:=(1+(x-1)q)D_{-1}=(1+(x-1)q)(1+(q-1)x)\partial.$

######  Proposition 2.2.

The operators $D_{0}$ and $S_{q}$ anti-commute, where $S_{q}$ acts on the
space of functions by precomposition.

The proof is a direct verification, similar to the proof of Proposition 2.1.
An equivalent statement is $S_{q}\circ D_{0}\circ S_{q}=-D_{0}$.

###### Proof.

For a function $f(x)$, we have on the one side

$D_{0}\circ
S_{q}(f(x))=D_{0}f\left(-\frac{1}{qx}\right)=(1+(x-1)q)(1+(q-1)x)f^{\prime}\left(-\frac{1}{qx}\right)\frac{1}{qx^{2}}.$

On the other hand,

$\displaystyle S_{q}\circ D_{0}(f(x))$
$\displaystyle=S_{q}((1+(x-1)q)(1+(q-1)x)f^{\prime}(x))$
$\displaystyle=\left(1+q\left(-\frac{1}{qx}-1\right)\right)\left(1-\frac{1}{qx}(q-1)\right)f^{\prime}\left(-\frac{1}{qx}\right)$
$\displaystyle=-\frac{1}{qx^{2}}(1+(x-1)q)(1+(q-1)x)f^{\prime}\left(-\frac{1}{qx}\right).$
∎

More generally, we can find all operators $D$ of the form $p(x)\partial$ which
anti-commute with $S_{q}$. The relation $\\{D,S_{q}\\}=0$ gives

$p(x)=-qx^{2}p\left(-\frac{1}{qx}\right).$

Adding as constraint that $p$ has to be polynomial, it is clear that it is of
degree at most 2. Plugging in $p(x)=p_{0}+p_{1}x+p_{2}x^{2}$ gives a solution
for any $p_{1}$ and $p_{2}=-qp_{0}$. In other words, the two fundamental
solutions are $p(x)=x$ and $p(x)=1-qx^{2}$. Note in particular that the
undeformed operator $x\partial$ still anticommutes with $S_{q}$. The
particular choice above for $D_{0}$ is ${p_{1}=-1+3q-q^{2}}$ and $p_{0}=1-q$.
We will see below why this is the simplest choice.

Let us determine the eigenfunctions of $D_{0}$ with eigenvalue $\alpha$. One
has to solve $\alpha f=D_{0}f$, i.e., $(\ln
f)^{\prime}=\frac{\alpha}{(1+(q-1)x)(1+(x-1)q)}$. The solutions are

$\left(\frac{1+(x-1)q}{1+(q-1)x}\right)^{\frac{\alpha}{q^{2}-q+1}}.$

We define the _$q$ -rational transition map_

$g_{q}(x)=\frac{1+(x-1)q}{1+(q-1)x},$ (2.1)

which is the unique eigenfunction of $D_{0}$ with eigenvalue $q^{2}-q+1$ and
normalization $g_{q}(0)=1-q$. We can think of $g_{q}$ as a deformation of the
identity map. We study this function more in detail below in Section 2.2.

Now we come back to the discussion why our $D_{0}$ is the simplest choice.
Consider an operator ${D=p(x)\partial}$ anti-commuting with $S_{q}$, i.e., of
the form $p(x)=p_{0}+p_{1}x-qp_{0}x^{2}$ with arbitrary
${p_{0},p_{1}\in\mathbb{Z}[q]}$. We impose that $D$ deforms $x\partial$, that
is $p_{0}(1)=0$ and $p_{1}(1)=1$. We also impose the leading terms of $p_{0}$,
$p_{1}$ to be $\pm 1$. We wish that the eigenfunctions of $D$ are Möbius
transformations in $\mathbb{Z}[q]$. This is only the case if the discriminant
of $p_{0}+p_{1}x-qp_{0}x^{2}$ is a square in $\mathbb{Z}[q]$. This leads to
the equation $p_{1}(q)^{2}+4qp_{0}(q)^{2}=R(q)^{2}$ for some
$R\in\mathbb{Z}[q]$. This is equivalent to $4qp_{0}^{2}=(R-p_{1})(R+p_{1})$.
Excluding the case where $p_{0}=0$ which leads to the undeformed operator
$x\partial$, the next simplest case is $p_{0}(q)=1-q$. By treating all
possible factorizations of $4q(1-q)^{2}$, we see that the $p_{1}$ with lowest
degree has to be $p_{1}(q)=-1+3q-q^{2}$ which is the case for our choice
$D_{0}$.

We complete the operators $D_{-1}$ and $D_{0}$ to a deformed
$\mathfrak{sl}_{2}$. For that, we wish to deform $x^{2}\partial$. Note that
$x^{2}\partial=S\circ\partial\circ S$. This motivates the following
definition:

$D_{1}:=S_{q}\circ D_{-1}\circ S_{q}=(1+(x-1)q)x\partial.$

By definition, $D_{1}$ commutes with $S_{q}T_{q}S_{q}$.

Our first result is that these three operators give a Lie algebra deforming
$\mathfrak{sl}_{2}$:

######  Theorem 2.3.

The operators $D_{-1}$, $D_{0}$ and $D_{1}$ form a Lie algebra with brackets

$\displaystyle[D_{0},D_{1}]=\big{(}q^{2}-q+1\big{)}D_{1}+(1-q)D_{0},\qquad[D_{0},D_{-1}]=-\big{(}q^{2}-q+1\big{)}D_{-1}+(1-q)D_{0},$
$\displaystyle[D_{-1},D_{1}]=2D_{0}+(1-q)(D_{1}-D_{-1}).$

For $q=1$, we get the Lie algebra $\mathfrak{sl}_{2}$.

###### Proof.

The proof is a straightforward computation. All $D_{i}$ are of the form
$g(x)\partial$ with $g$ a polynomial of degree at most 2. This explains why we
can express any bracket as linear combination of $D_{-1}$, $D_{0}$ and
$D_{1}$. The non-trivial part is that the coefficients are in $\mathbb{Z}[q]$.
Since ${D_{0}=(1+(x-1)q)D_{-1}}$, we get

$[D_{0},D_{-1}]=-D_{-1}(1+(x-1)q)D_{-1}=-q(1+(q-1)x)^{2}\partial.$

Similarly, we have $D_{0}=(1+(q-1)x)x^{-1}D_{1}$, hence

$[D_{0},D_{1}]=-D_{1}\big{(}x^{-1}+q-1\big{)}D_{1}=(1+(x-1)q)^{2}\partial.$

The last bracket can be computed to be
$[D_{-1},D_{1}]=\big{(}1-q+2qx+q(q-1)x^{2}\big{)}\partial$. One explicitely
checks that these three brackets coincide with results claimed in the theorem.

Finally, it is clear that these brackets satisfy the Jacobi identity since we
know a representation of the operators $D_{i}$ as differential operators. ∎

The Lie algebra $\mathfrak{sl}_{2}$ being simple, it does not allow any non-
trivial deformations. Our $q$-deformation is indeed abstractly isomorphic to
$\mathfrak{sl}_{2}$ when $q$ and $q^{2}-q+1$ are invertible. To give an
explicit isomorphism, denote by $(f,h,e)$ the generators of
$\mathfrak{sl}_{2}$ given by the differential operators
$\big{(}\partial,x\partial,x^{2}\partial\big{)}$. They satisfy $[h,e]=e$,
$[h,f]=-f$ and $[e,f]=-2h$. The following is an isomorphism of Lie algebras
between $(D_{-1},D_{0},D_{1})$ and $(f,h,e)$:

$\displaystyle f=q^{-1/2}\left(D_{-1}+\frac{q-1}{q^{2}-q+1}D_{0}\right),\qquad
h=\frac{D_{0}}{q^{2}-q+1},$ $\displaystyle
e=q^{-1/2}\left(D_{1}+\frac{1-q}{q^{2}-q+1}D_{0}\right).$

Using this isomorphism to $\mathfrak{sl}_{2}$, we can describe a 2-dimensional
representation of the deformed Lie algebra defined by $(D_{-1},D_{0},D_{1})$.
Using the standard realization $f=\left(\begin{smallmatrix}0&0\\\
1&0\end{smallmatrix}\right)$, $h=\left(\begin{smallmatrix}1/2&0\\\
0&-1/2\end{smallmatrix}\right)$ and $e=\left(\begin{smallmatrix}0&-1\\\
0&0\end{smallmatrix}\right)$, we get

$D_{-1}=\begin{pmatrix}\tfrac{1-q}{2}&0\vspace{1mm}\\\
q^{1/2}&\tfrac{q-1}{2}\end{pmatrix},\qquad
D_{0}=\begin{pmatrix}\tfrac{q^{2}-q+1}{2}&0\\\
0&\tfrac{-q^{2}+q-1}{2}\end{pmatrix},\qquad
D_{1}=\begin{pmatrix}\tfrac{q-1}{2}&-q^{1/2}\\\
0&\tfrac{1-q}{2}\end{pmatrix}.$

Note that this representation is not in $\mathfrak{sl}_{2}(\mathbb{Q}[q])$. A
direct computation shows that there is no 2-dimensional representation of our
$q$-deformed $\mathfrak{sl}_{2}$ into $\mathfrak{sl}_{2}(\mathbb{Q}[q])$. In
dimension 3, there is of course the adjoint representation into
$\mathfrak{sl}_{3}(\mathbb{Z}[q])$.

######  Remark 2.4.

It is tempting to consider $D_{-1}$, $D_{1}$ and
$\widehat{D}_{0}:=[D_{-1},D_{1}]$. The operator $\widehat{D}_{0}$ still anti-
commutes with $S_{q}$ and the bracket relations are

$[\widehat{D}_{0},D_{\pm 1}]=\pm\big{(}q^{2}+1\big{)}D_{\pm
1}\pm(q-1)^{2}D_{\mp 1}.$

The main drawback of this choice is that the eigenfunctions of
$\widehat{D}_{0}$ are Möbius transformations with coefficients not in
$\mathbb{Z}[q]$.

######  Remark 2.5.

A simpler and very similar Lie algebra deforming $\mathfrak{sl}_{2}$ is given
by generators $(d_{-1},d_{0},d_{1})$ with brackets

$\displaystyle[d_{0},d_{-1}]=-qd_{-1}+(1-q)d_{0},\qquad[d_{0},d_{1}]=qd_{1}+(1-q)d_{0},$
$\displaystyle[d_{-1},d_{1}]=2d_{0}+(1-q)(d_{1}-d_{-1}).$

It can be obtained as our deformation for a formal parameter $q$ with relation
$(q-1)^{2}=0$. Then ${q^{2}-q+1=q}$. One checks that the Jacobi identity still
holds.

###  2.2 $\boldsymbol{q}$-rational transition map

The map $g_{q}$ defined in (2.1) plays a fundamental role, both for
generalizing the $q$-deformation from $\mathfrak{sl}_{2}$ to the Witt algebra
in Section 3, and in the theory of $q$-deformed rationals as we shall see now.
It allows to pass between two different $q$-deformations of the rational
numbers.

Recall that the $q$-rational transition map is defined by

$g_{q}(x)=\frac{1+(x-1)q}{1+(q-1)x},$

which is a deformation of the identity. It is the eigenfunction of $D_{0}$
with eigenvalue $q^{2}-q+1$ and normalization $g_{q}(0)=1-q$. Note that
$g_{q}$ is a Möbius transformation associated to the matrix

$\begin{pmatrix}q&1-q\\\ q-1&1\end{pmatrix},$

which is of determinant $q^{2}-q+1$. For $q\neq 1$, $g_{q}$ is an elliptic
transformation since its normalized trace is given by

$\frac{q+1}{\sqrt{q^{2}-q+1}}<2.$

The unique fixed point on $\mathbb{H}^{2}$ is $\tfrac{1+{\rm i}\sqrt{3}}{2}$
which is independent of $q$.

From the definition of $g_{q}$, we see the following duality between $q$ and
$x$:

$g_{q}(x)g_{x}(q)=1.$

######  Proposition 2.6.

The functions $g_{q}$, $T_{q}$ and $S_{q}$, seen as $2\times 2$ matrices
satisfy:

$g_{q}T_{q}=qT_{q^{-1}}g_{q}\qquad\text{and}\qquad
g_{q}S_{q}=qS_{q^{-1}}g_{q}.$

Therefore, seen as Möbius transformations, we have $g_{q}\circ
T_{q}=T_{q^{-1}}\circ g_{q}$ and $g_{q}\circ S_{q}=S_{q^{-1}}\circ g_{q}$.

###### Proof.

Both assertions can be checked by a direct computation:

$g_{q}T_{q}=\begin{pmatrix}q&1-q\\\ q-1&1\end{pmatrix}\begin{pmatrix}q&1\\\
0&1\end{pmatrix}=\begin{pmatrix}q^{2}&1\\\
q^{2}-q&q\end{pmatrix}=qT_{q^{-1}}g_{q},$

and similarly

$g_{q}S_{q}=\begin{pmatrix}q&1-q\\\ q-1&1\end{pmatrix}\begin{pmatrix}0&-1\\\
q&0\end{pmatrix}=\begin{pmatrix}q-q^{2}&-q\\\
q&1-q\end{pmatrix}=qS_{q^{-1}}g_{q}.$

The second identity can be derived also as follows: since
$S_{q}D_{0}S_{q}=-D_{0}$, we see that both $g_{q}^{-1}(x)$ and
$g_{q}(S_{q}(x))$ are eigenfunctions of $D_{0}$ with eigenvalue $-q^{2}+q-1$.
Hence they have to be multiple of each other. The precise relation is given by
$g_{q}(S_{q}(x))=\tfrac{-q}{g_{q}(x)}=S_{q^{-1}}(g_{q}(x))$. ∎

We describe now the main link to $q$-deformed rational numbers. In [13, Remark
3.2], the authors notice that the procedure for $q$-deformed irrational
numbers gives two different answers when applied to rationals. This was
further developed in [2], from which we borrow the notations. When one
approaches a rational $r/s$ from the right by a sequence of rationals strictly
bigger than $r/s$, the procedure gives the so-called _right $q$-rational_
$[r/s]_{q}^{\sharp}$. This is the deformation obtained from applying $T_{q}$
and $S_{q}$ to zero described at the beginning of the Introduction. When
approaching $r/s$ from the left, the limit gives another $q$-deformation of
$r/s$, called _left $q$-rational_ and denoted by $[r/s]_{q}^{\flat}$ [2,
Theorem 2.11].

The precise formulas given in [2, Definition 2.6] can be written in our
context as follows: consider $U=TST$, which is the function
$U(x)=\tfrac{1}{1+1/x}$, and its $q$-analog $U_{q}=T_{q}S_{q}T_{q}$. For a
rational $r/s\in\mathbb{Q}$, take the unique even continued fraction
expression $r/s=[a_{1},a_{2},\dots,a_{2n}]$. This means that
$r/s=T^{a_{1}}U^{a_{2}}T^{a_{3}}\cdots U^{a_{2n}}(\infty)$. By convention, we
put $\infty=[\,]$, the empty expression. Then

$\left[\frac{r}{s}\right]_{q}^{\sharp}=T_{q}^{a_{1}}U_{q}^{a_{2}}T_{q}^{a_{3}}\cdots
U_{q}^{a_{2n}}(\infty),$ (2.2)

and

$\left[\frac{r}{s}\right]_{q}^{\flat}=T_{q}^{a_{1}}U_{q}^{a_{2}}T_{q}^{a_{3}}\cdots
U_{q}^{a_{2n}}\left(\frac{1}{1-q}\right).$ (2.3)

To give some examples, we have $[0]_{q}^{\sharp}=0$ and
$[0]_{q}^{\flat}=\tfrac{q-1}{q}$, $[1]_{q}^{\sharp}=1$ and
$[1]_{q}^{\flat}=q$, $[2]_{q}^{\sharp}=1+q$ and $[2]_{q}^{\flat}=1+q^{2}$,
$[\infty]_{q}^{\sharp}=\infty$ and $[\infty]_{q}^{\flat}=\tfrac{1}{1-q}$.

It was noticed numerically by Valentin Ovsienko that $g_{q}$ is a transition
between these two $q$-deformations of rational numbers. This is made precise
in the following:

######  Theorem 2.7.

The passage between the two $q$-deformations of rationals is given by

$g_{q}\left(\left[\frac{r}{s}\right]_{q}^{\sharp}\right)=\left[\frac{r}{s}\right]_{q^{-1}}^{\flat}.$

Note that $q$ gets inversed to $q^{-1}$. The proof is an application of
Proposition 2.6.

###### Proof.

Proposition 2.6 gives $g_{q}U_{q}=U_{q^{-1}}g_{q}$. Using equation (2.2) and
again Proposition 2.6, we get

$g_{q}\left(\left[\frac{r}{s}\right]_{q}^{\sharp}\right)=g_{q}T_{q}^{a_{1}}U_{q}^{a_{2}}T_{q}^{a_{3}}\cdots
U_{q}^{a_{2n}}(\infty)=T_{q^{-1}}^{a_{1}}U_{q^{-1}}^{a_{2}}T_{q^{-1}}^{a_{3}}\cdots
U_{q^{-1}}^{a_{2n}}g_{q}(\infty).$

Now $g_{q}(\infty)=\tfrac{q}{q-1}=\tfrac{1}{1-q^{-1}}$. Hence we conclude by
equation (2.3). ∎

As an application, we can reprove the positivity property of left
$q$-rationals, proven in [2, Appendix A.1] via an explicit combinatorial
interpretation.

######  Corollary 2.8.

For $r/s>1$, we have $R^{\flat},S^{\flat}\in\mathbb{N}[q]$, where
$[r/s]_{q}^{\flat}=R^{\flat}(q)/S^{\flat}(q)$.

###### Proof.

From [12, Proposition 1.3], we know that for $r/s\in\mathbb{Q}_{>1}$ the right
$q$-rational $[r/s]_{q}^{\sharp}=R^{\sharp}(q)/S^{\sharp}(q)$ is a rational
function in $q$ with positive coefficients, i.e.,
$R^{\sharp},S^{\sharp}\in\mathbb{N}[q]$. We also know from [12, Theorem 2]
that if $r/s>r^{\prime}/s^{\prime}$, then
$R^{\sharp}S^{\prime\sharp}-R^{\prime\sharp}S^{\sharp}\in\mathbb{N}[q]$. Since
${r/s>1}$, we get ${R^{\sharp}-S^{\sharp}\in\mathbb{N}[q]}$. Since $r/s+1>r/s$
and $[r/s+1]_{q}^{\sharp}=q[r/s]_{q}^{\sharp}+1$, we also get
${(q-1)R^{\sharp}+S^{\sharp}\in\mathbb{N}[q]}$. Finally, we deduce that

$\left[\frac{r}{s}\right]_{q^{-1}}^{\flat}=g_{q}\left(\left[\frac{r}{s}\right]_{q}^{\sharp}\right)=\frac{q\big{(}R^{\sharp}-S^{\sharp}\big{)}+S^{\sharp}}{(q-1)R^{\sharp}+S^{\sharp}}$

has positive coefficients. Multiplying both numerator and denominator with an
appropriate power of $q$, we get the same for $[r/s]^{\flat}_{q}$. ∎

Finally, we can use the transition function $g_{q}(x)$ as reparametrization of
$\mathbb{P}^{1}$. To emphasize the dependence of our differential operators,
we will write here $D_{-1}(q,x)=(1+(q-1)x)\partial_{x}$ and similar for
$D_{0}$ and $D_{1}$.

######  Proposition 2.9.

Reparametrizing $\mathbb{P}^{1}$ by the transition map $\xi=g_{q}(x)$ gives

$\displaystyle D_{\pm 1}(q,x)$ $\displaystyle=qD_{\pm
1}\big{(}q^{-1},\xi\big{)},\qquad
D_{0}(q,x)=\big{(}q^{2}-q+1\big{)}\xi\partial_{\xi}.$

The behavior of $D_{-1}$ and $D_{1}$ is reminiscent of Proposition 2.6.

###### Proof.

Using $\frac{{\rm d}\xi}{{\rm d}x}=\tfrac{q^{2}-q+1}{(1+(q-1)x)^{2}}$ and
$x=\tfrac{\xi+q-1}{q+(1-q)\xi}$, we get
$1+(q-1)x=\tfrac{q^{2}-q+1}{q+(1-q)\xi}$. Hence

$D_{-1}(x,q)=(1+(q-1)x)\frac{{\rm d}\xi}{{\rm
d}x}\partial_{\xi}=\frac{q^{2}-q+1}{1+(q-1)x}\partial_{\xi}=(q+(1-q)\xi)\partial_{\xi}=qD_{-1}\big{(}q^{-1},\xi\big{)}.$

The computation for $D_{1}$ is similar. Finally,

$D_{0}(q,x)=(1+(q-1)x)(1+(x-1)q)\partial_{x}=\big{(}q^{2}-q+1\big{)}g_{q}(x)\partial_{\xi}=\big{(}q^{2}-q+1\big{)}\xi\partial_{\xi}.$

This proposition indicates that we can use the undeformed operator $x\partial$
together with $D_{\pm 1}$ to get a deformation of $\mathfrak{sl}_{2}$ which is
equivalent to our proposal. The importance of the $q$-rational transition map
$g_{q}$, especially in the light of Proposition 2.6, justifies to use $D_{0}$
instead of $x\partial$.

###  2.3 Heisenberg algebra

The operators $D_{-1}$ and $g_{q}$, seen as multiplication operator, give a
deformation of the Heisenberg algebra. This strengthens the idea of
considering $g_{q}$ as a deformation of the identity.

######  Theorem 2.10.

The two operators $D_{-1}$ and $g_{q}$ satisfy

$[D_{-1},g_{q}]=q+(1-q)g_{q}.$

Hence together with the central element $1$, they define a solvable
$3$-dimensional Lie algebra deforming the $3$-dimensional Heisenberg algebra
which we recover for $q=1$.

The proof is a simple computation:

$[D_{-1},g_{q}]=D_{-1}(g_{q})=\frac{q^{2}-q+1}{1+(q-1)x}=q+(1-q)g_{q}.$

The Lie algebra generated by $(1,g_{q},D_{-1})$ is solvable since $D_{-1}$ is
not in the image of the Lie bracket. Hence the derived series becomes zero at
the second step.

The previous theorem works since there is a nice expression for
$D_{-1}(g_{q})$. This holds true more generally:

######  Proposition 2.11.

The function $g_{q}$ behaves well under the operators $D_{-1}$, $D_{0}$ and
$D_{1}$:

$\displaystyle D_{0}(g_{q})$
$\displaystyle=\big{(}q^{2}-q+1\big{)}g_{q},\qquad
D_{-1}(g_{q})=q+(1-q)g_{q},\qquad D_{1}(g_{q})=(q-1)g_{q}+g_{q}^{2}.$

###### Proof.

We only have to prove the last statement since we have already seen the first
two. For that, we use the relation $g_{q}(S_{q}(x))=-q/g_{q}(x)$, see
Proposition 2.6. We get

$D_{1}(g_{q})=S_{q}D_{-1}S_{q}(g_{q})=S_{q}D_{-1}(-q/g_{q})=S_{q}\bigl{(}-qg_{q}^{-2}(q+(1-q)g_{q})\bigr{)},$

where we first used that $S_{q}$ acts by precomposition, then Proposition 2.6
and finally the expression for $D_{-1}(g_{q})$. Since $S_{q}$ acts by
precomposition applying 2.6 again concludes:

$D_{1}(g_{q})S_{q}\bigl{(}-qg_{q}^{-2}(q+(1-q)g_{q})\bigr{)}=g_{q}^{2}+(q-1)g_{q}.$

We can use this proposition to express one operator in terms of another via
the relation $D_{i}=\tfrac{D_{i}(g_{q})}{D_{j}(g_{q})}D_{j}$ for all
$i,j\in\\{-1,0,1\\}$. This holds true since these differential operators are
of order 1.

##  3 Deformed Witt algebra

Now that we have deformed the differential operators $\partial$, $x\partial$
and $x^{2}\partial$, we can do the same for all $x^{n}\partial$ for
$n\in\mathbb{Z}$. These are a realization of the _Witt algebra_ , the Lie
algebra of complex polynomial vector fields on the circle (the centerless
Virasoro algebra). Putting $\ell_{n}=x^{n+1}\partial$, the Lie algebra
structure is given by

$[\ell_{n},\ell_{m}]=(m-n)\ell_{n+m}.$

To get a deformation of the Witt algebra, we define for $n>1$:

$\displaystyle D_{n}$ $\displaystyle=g_{q}^{n-1}D_{1},\qquad
D_{-n}=\big{(}qg_{q}^{-1}\big{)}^{n-1}D_{-1},$

where $g_{q}^{-1}=1/g_{q}$ denotes the inverse for multiplication (not
composition).

######  Proposition 3.1.

The operators $D_{n}$ behave nicely when multiplied by $g_{q}$. By definition
we have ${g_{q}D_{n}=D_{n+1}}$ for $n\geq 1$ and $g_{q}D_{-n}=qD_{-n+1}$ for
$n\geq 2$. In addition,

$\displaystyle g_{q}D_{0}$
$\displaystyle=(1-q)D_{0}+\big{(}q^{2}-q+1\big{)}D_{1},\qquad
g_{q}D_{-1}=D_{0}+(1-q)D_{1}.$

Similarly, there is a nice behavior when multiplied by $qg_{q}^{-1}$. By
definition $qg_{q}^{-1}D_{-n}=D_{-n-1}$ for ${n\geq 1}$ and
$qg_{q}^{-1}D_{n}=qD_{n-1}$ for $n\geq 2$. In addition,

$\displaystyle qg_{q}^{-1}D_{0}$
$\displaystyle=(q-1)D_{0}+\big{(}q^{2}-q+1\big{)}D_{-1},\qquad
qg_{q}^{-1}D_{1}=D_{0}+(q-1)D_{-1}.$

###### Proof.

From the definitions, we get $g_{q}D_{0}=(1+(x-1)q)^{2}\partial$. From
Proposition 2.3 and its proof, we see that this is $[D_{0},D_{1}]$. Therefore,
$g_{q}D_{0}=\big{(}q^{2}-q+1\big{)}D_{1}+(1-q)D_{0}$. A direct computation
also gives $g_{q}D_{-1}=(1+(x-1)q)\partial=D_{0}+(1-q)D_{1}$.

For the second half, note that

$g_{q}D_{0}+(q-1)g_{q}D_{-1}=\big{(}q^{2}-q+1-(q-1)^{2}\big{)}D_{1}=qD_{1}.$

Dividing by $g_{q}$ gives $qg_{q}^{-1}D_{1}=D_{0}+(q-1)D_{-1}$. Similarly,

$(q-1)g_{q}D_{0}+\big{(}q^{2}-q+1\big{)}D_{-1}=qD_{0},$

so dividing by $g_{q}$ gives
$qg_{q}^{-1}D_{0}=(q-1)D_{0}+\big{(}q^{2}-q+1\big{)}D_{-1}$. ∎

Using Propositions 2.11 and 3.1, we get the bracket relations of all $D_{n}$:

######  Theorem 3.2.

The $(D_{n})_{n\in\mathbb{Z}}$ form a Lie algebra with bracket given by
$($with $n,r>0)$:

$\displaystyle[D_{0},D_{n}]=n\big{(}q^{2}-q+1\big{)}D_{n}+\big{(}q^{2}-q+1\big{)}\displaystyle\sum_{k=1}^{n-1}(1-q)^{k}D_{n-k}+(1-q)^{n}D_{0},$
$\displaystyle[D_{0},D_{-n}]=-n\big{(}q^{2}-q+1\big{)}D_{-n}-\big{(}q^{2}-q+1\big{)}\displaystyle\sum_{k=1}^{n-1}(q-1)^{k}D_{-n+k}-(q-1)^{n}D_{0},$
$\displaystyle[D_{n},D_{n+r}]=rD_{2n+r}+(q-1)rD_{2n+r-1},$
$\displaystyle[D_{-n},D_{-n-r}]=-rD_{-2n-r}+(q-1)rD_{-2n-r+1},$
$\displaystyle[D_{-n},D_{n}]=2nq^{n-1}D_{0}+(2n-1)q^{n-1}(q-1)(D_{-1}-D_{1}),$
$\displaystyle[D_{n+r},D_{-n}]=(q-1)q^{n-1}(2n+r-1)D_{r+1}-\big{(}q^{2}+(2n+r-2)q+1\big{)}q^{n-1}D_{r},$
$\displaystyle\phantom{[D_{n+r},D_{-n}]=}{}-q^{n-1}\big{(}q^{2}-q+1\big{)}\displaystyle\sum_{k=1}^{r-1}(1-q)^{k}D_{r-k}-(1-q)^{r}q^{n-1}D_{0},$
$\displaystyle[D_{n},D_{-n-r}]=-(q-1)q^{n-1}(2n+r-1)D_{-r-1}-\big{(}q^{2}+(2n+r-2)q+1\big{)}q^{n-1}D_{-r},$
$\displaystyle\phantom{[D_{n},D_{-n-r}]=}{}-q^{n-1}\big{(}q^{2}-q+1\big{)}\displaystyle\sum_{k=1}^{r-1}(q-1)^{k}D_{-r+k}-(q-1)^{r}q^{n-1}D_{0}.$

For $q=1$, one recovers the Witt algebra.

It is clear that the bracket of the operators $D_{n}$ satisfies the Jacobi
identity since these operators come from a realization as differential
operators. We only have to check the bracket relations, which uses induction
and all properties between $g_{q}$ and $D_{-1}$, $D_{0}$, $D_{1}$.

###### Proof.

We prove the first relation by induction on $n$. The case $n=1$ is true by
Proposition 2.3. Then for $n>1$,

$\displaystyle[D_{0},D_{n}]={}$
$\displaystyle[D_{0},g_{q}D_{n-1}]=D_{0}(g_{q})D_{n-1}+g_{q}[D_{0},D_{n-1}]$
$\displaystyle={}$
$\displaystyle\big{(}q^{2}-q+1\big{)}g_{q}D_{n-1}+g_{q}(n-1)\big{(}q^{2}-q+1\big{)}D_{n-1}$
$\displaystyle+g_{q}\left(\big{(}q^{2}-q+1\big{)}\displaystyle\sum_{k=1}^{n-2}(1-q)^{k}D_{n-1-k}+(1-q)^{n-1}D_{0}\right)$
$\displaystyle={}$ $\displaystyle
n\big{(}q^{2}-q+1\big{)}D_{n}+\big{(}q^{2}-q+1\big{)}\displaystyle\sum_{k=1}^{n-1}(1-q)^{k}D_{n-k}+(1-q)^{n}D_{0},$

where we used that $g_{q}$ is an eigenfunction of $D_{0}$, and Proposition
3.1. The second statement is a similar computation. The third relation comes
as follows:

$\displaystyle[D_{n},D_{n+r}]$
$\displaystyle=[D_{n},g_{q}^{r}D_{n}]=D_{n}(g_{q}^{r})D_{n}=rg_{q}^{r-1}g_{q}^{n-1}D_{1}(g_{q})D_{n}$
$\displaystyle=rg_{q}^{r+n-2}\big{(}g_{q}^{2}+(q-1)g_{q}\big{)}D_{n}=rD_{2n+r}+r(q-1)D_{2n+r-1},$

where we used Proposition 2.11 for $D_{1}(g_{q})$. The fourth bracket is a
similar computation. To prove the fifth relation, we use induction on $n$
again. The initial $n=1$ is done by Proposition 2.3. Then for $n\geq 1$,

$\displaystyle[D_{-n-1},D_{n+1}]={}$
$\displaystyle\big{[}qg_{q}^{-1}D_{-n},g_{q}D_{n}\big{]}=qg_{q}^{-1}D_{-n}(g_{q})D_{n}+q[D_{-n},D_{n}]-qg_{q}D_{n}\big{(}g_{q}^{-1}\big{)}D_{-n}$
$\displaystyle={}$ $\displaystyle
q^{n}g_{q}^{-n}(q+(1-q)g_{q})D_{n}+q\big{(}2nq^{n-1}D_{0}$
$\displaystyle+(2n-1)q^{n-1}(q-1)(D_{-1}-D_{1})\big{)}+qg_{q}^{n-2}\big{(}g_{q}^{2}+(q-1)g_{q}\big{)}D_{-n}$
$\displaystyle={}$ $\displaystyle
2nq^{n}D_{0}+q^{n}(2n(q-1)+q_{q})D_{-1}-q^{n}\big{(}2n(q-1)-qg_{q}^{-1}\big{)}D_{1}$
$\displaystyle={}$
$\displaystyle(2n+2)q^{n}D_{0}+(2n+1)q^{n}(q-1)(D_{-1}-D_{1}),$

where we used several times Propositions 2.11 and 3.1. Finally, for the last
two brackets, we start from (where $a,b>0$)

$\displaystyle[D_{a},D_{-b}]$
$\displaystyle=\big{[}g_{q}^{a-1}D_{1},\big{(}qg_{q}^{-1}\big{)}^{b-1}D_{-1}\big{]}$
$\displaystyle=q^{b-1}\big{(}g_{q}^{a-1}D_{1}\big{(}g_{q}^{1-b}\big{)}D_{-1}-g_{q}^{1-b}D_{-1}\big{(}g_{q}^{a-1}\big{)}D_{1}+g_{q}^{a-b}[D_{1},D_{-1}]\big{)}$
$\displaystyle=q^{b-1}g_{q}^{a-b}(-(a+b)D_{0}+(a+b-1)(q-1)(D_{1}-D_{-1})).$
(3.1)

An easy induction gives for $r>0$,

$g_{q}^{r}D_{0}=\big{(}q^{2}-q+1\big{)}\sum_{k=0}^{r-1}(1-q)^{k}D_{r-k}+(1-q)^{r}D_{0}.$
(3.2)

Also we get $g_{q}^{r}D_{-1}=g_{q}^{r-1}D_{0}+(1-q)D_{r}$, where we can use
equation (3.2) to express the first term. Similar results hold for
$\big{(}qg_{q}^{-1}\big{)}^{r}D_{0}$ and $\big{(}qg_{q}^{-1}\big{)}^{r}D_{1}$.
Putting $a=n+r$ and $b=n$ in equation (3.1) and using (3.2) gives the bracket
$[D_{n+r},D_{-n}]$. Putting $a=n$ and $b=n+r$ gives in a similar way the last
bracket $[D_{n},D_{-n-r}]$. ∎

######  Remark 3.3.

Regarding Remark 2.5, we could try to simplify the defining relations of the
$q$-deformed Witt algebra by considering a formal parameter $q$ satisfying
$(q-1)^{2}=0$ (and then forget about this relation again). In contrast to the
$q$-deformed $\mathfrak{sl}_{2}$, the result here is not a Lie algebra
anymore. The Jacobi identity does not hold exactly, but only modulo
$(q-1)^{2}=0$.

##  4 Möbius transformations

The differential operators $\partial$, $x\partial$, $x^{2}\partial$ can be
interpreted in at least three different ways: first as differential operators
on $\mathbb{P}^{1}$ written in one chart (this was our approach). Second they
can be seen as complex vector fields on the circle
$\mathbb{S}^{1}\subset\mathbb{C}$ (this approach was used for the Witt
algebra). Third, a Lie algebra can be realised as Killing vector fields on the
associated symmetric space of non-compact type. For $\mathfrak{sl}_{2}$ this
symmetric space is the hyperbolic plane $\mathbb{H}^{2}$.

In this section, we integrate the operators $D_{-1}$, $D_{0}$ and $D_{1}$ seen
as vector fields of $\mathbb{H}^{2}$. The result gives interesting Möbius
transformations with $q$-parameter. In the Taylor expansion around ${q\to 1}$
(the “semi-classical limit”), we recover the deformed translation $T_{q}$.
Conjecturally there should be a $q$-deformation of $\mathbb{H}^{2}$ on which
these transformations act, such that the boundary can be identified with the
$q$-deformed real numbers of [13].

###  4.1 Classical setting

Consider first the classical setup with the operators $\partial$, $x\partial$
and $x^{2}\partial$. These are Killing vector fields on the hyperbolic plane
$\mathbb{H}^{2}$, whose integration determines isometries of $\mathbb{H}^{2}$.
Here, we consider ${\mathbb{H}^{2}\subset\mathbb{C}}$ in the upper half-plane
model and use the coordinate $x\in\mathbb{C}$.

To start, consider the case of the vector field
$V=\partial=\tfrac{\partial}{\partial x}$. A curve $\gamma$ integrates this
vector field iff $\gamma^{\prime}(t)=V(\gamma(t))=1$ for all $t\in\mathbb{R}$
(where we identified 1 with the constant vector field $\partial$). With
initial condition $\gamma(0)=x$ we get $\gamma(t)=x+t$. We should think of
this as a function $\gamma_{x}(t)$ of the initial condition. For time 1, we
get the translation $\gamma_{x}(1)=x+1=T(x)$.

Another important case is $x\partial$, for which we have to solve
$\gamma^{\prime}(t)=\gamma(t)$. With initial condition ${\gamma(0)=x}$, we
obviously get $\gamma(t)={\rm e}^{t}x$. The function $x\mapsto{\rm e}^{t}x$ is
the hyperbolic isometry of $\mathbb{H}^{2}$ associated to the geodesic joining
0 to $\infty$. Its matrix is given by

$\begin{pmatrix}{\rm e}^{t/2}&0\\\ 0&{\rm e}^{-t/2}\end{pmatrix}.$ (4.1)

We can immediately generalize to the generators of the Witt algebra. Consider
the operator $x^{n}\partial$ with $n\in\mathbb{Z}$, $n\neq 1$. A curve
$\gamma$ integrates the associated vector field if
$\gamma^{\prime}(t)=\gamma(t)^{n}$. The solution with initial condition
$\gamma(0)=x$ is given by

$\gamma_{x}(t)=\frac{x}{\big{(}1-(n-1)tx^{n-1}\big{)}^{1/(n-1)}}.$

Apart from $n=0$ and $n=2$, the associated transformations in $x$ are not
Möbius transformations. We get Möbius transformations though when passing to a
ramified covering. Putting $y=x^{n-1}$, we get

$\gamma_{x}(t)^{n-1}=\frac{y}{1-(n-1)ty}.$

###  4.2 Deformed transformations

We repeat the method of the previous subsection to deduce the transformations
associated to $D_{-1}$, $D_{0}$ and $D_{1}$. Since these operators are still
of the form $p(x)\partial$ with $p$ a polynomial in $x$ of degree at most 2,
the vector fields $D_{i}$ are still Killing vector fields, so their
integration gives Möbius transformations.

Start with $D_{-1}\\!=\\!(1+(q-1)x)\partial$. The associated differential
equation is ${\gamma^{\prime}(t)=1+(q-1)\gamma(t)}$ with initial condition
$\gamma(0)=x$. Solving this equation is standard: first one solves the
homogeneous equation, then one uses the variation of the constant to finally
get

$\gamma(t)=-\frac{1}{q-1}+\left(x+\frac{1}{q-1}\right){\rm e}^{(q-1)t}.$

For $t=1$, we get the associated map
$x\mapsto-\tfrac{1}{q-1}+\big{(}x+\tfrac{1}{q-1}\big{)}{\rm e}^{q-1}$. The
Taylor expansion around $q-1$ at order 1 gives

$x\mapsto-\frac{1}{q-1}+\left(x+\frac{1}{q-1}\right)q=qx+1,$

which is nothing but $T_{q}(x)$. For a general time $t$, the same procedure
gives $x\mapsto(1-t+qt)x+t$. To sum up:

######  Proposition 4.1.

The time $1$ flow of the operator $D_{-1}$ seen as vector field on
$\mathbb{H}^{2}$ is the affine map
$x\mapsto-\tfrac{1}{q-1}+\big{(}x+\tfrac{1}{q-1}\big{)}{\rm e}^{q-1}$ whose
Taylor expansion at order $1$ in $q-1$ is $T_{q}$.

For the operator $D_{1}$, it is not necessary to do any computation since
$D_{1}=S_{q}D_{-1}S_{q}$. We can simply conjugate by $S_{q}$ the previous
computations. In particular, the associated transformation in the Taylor
expansion is $S_{q}T_{q}S_{q}$.

Consider now the operator $D_{0}=(1+(q-1)x)(1+(x-1)q)\partial$. The associated
differential equation reads

$\gamma^{\prime}(t)=1-q+\bigl{(}-1+3q-q^{2}\bigr{)}\gamma+q(q-1)\gamma^{2},$

which is a Ricatti equation.

To solve a Ricatti equation, put $a=1-q$, $b=-1+3q-q^{2}$, $c=q(q-1)$ and
introduce the new function $u$ such that $c\gamma(t)=-u^{\prime}(t)/u(t)$.
Then $u$ satisfies $u^{\prime\prime}(t)-bu^{\prime}(t)+acu(t)=0$. The
discriminant has the nice expression $b^{2}-4ac=\big{(}q^{2}-q+1\big{)}^{2}$.
The two roots of the characteristic equation are $q$ and $-(q-1)^{2}$. Hence
we get $u(t)=C_{1}{\rm e}^{qt}+C_{2}{\rm e}^{-(q-1)^{2}t}$, where $C_{1}$,
$C_{2}$ are two constants. Since $\gamma=-u^{\prime}/(cu)$, we can scale
$C_{1}$ and $C_{2}$ by the same number without changing $\gamma$. Putting
$C_{1}=1-q$, we get

$\gamma(t)=\frac{{\rm e}^{qt}+C_{2}\frac{q-1}{q}{\rm
e}^{-(q-1)^{2}t}}{(1-q){\rm e}^{qt}+C_{2}{\rm e}^{-(q-1)^{2}t}}.$

The initial condition $\gamma(0)=x$ gives

$C_{2}=\frac{q(1-q)x-q}{q-1-qx}.$

We already see that $\gamma_{x}(t)$ is a Möbius transformation in $x$ since
$C_{2}$ is. For time $t=1$, we get the following.

######  Proposition 4.2.

Integrating to time $t=1$ the operator $D_{0}$ seen as vector field in
$\mathbb{H}^{2}$ gives the Möbius transformation

$\gamma_{x}(1)=\frac{\big{(}q{\rm e}^{q}+(q-1)^{2}{\rm
e}^{-(q-1)^{2}}\big{)}x+(1-q)\big{(}{\rm e}^{q}-{\rm
e}^{-(q-1)^{2}}\big{)}}{q(1-q)\big{(}{\rm e}^{q}-{\rm
e}^{-(q-1)^{2}}\big{)}x+(1-q)^{2}{\rm e}^{q}+q{\rm e}^{-(q-1)^{2}}}.$

If we Taylor expand $\gamma_{x}(1)$ around $q-1$ to order 1, we get a
quadratic polynomial in $x$. In order to keep a Möbius transformation, we
Taylor expand all entries of the associated $2\times 2$ matrix to order 1 in
$q-1$. The result is

$W_{q}:=\begin{pmatrix}e(1-2q)&(e-1)(q-1)\\\ (e-1)(q-1)&-q\end{pmatrix},$

where we used $q^{2}=2q-1$ coming from the Taylor expansion. We see that $q=1$
gives the transformation $x\mapsto ex$.

A similar computation with arbitrary time $t$ gives

$W_{q}^{t}=\begin{pmatrix}{\rm e}^{t}(t-qt-q)&({\rm e}^{t}-1)(q-1)\\\ ({\rm
e}^{t}-1)(q-1)&-q\end{pmatrix},$

which for $q=1$ gives the transformation $x\mapsto{\rm e}^{t}x$ from equation
(4.1).

###  4.3 Speculations about a $\boldsymbol{q}$-deformed hyperbolic plane

The above computations seem to indicate the existence of a $q$-deformed
version of the hyperbolic plane $\mathbb{H}^{2}_{q}$ on which the
transformations $T_{q}$, $S_{q}$, $g_{q}$ and $W_{q}$ act. A similar idea is
developed in [2] where a compactification of the space of stability conditions
for type $A_{2}$ is constructed.

The transformation $S_{q}(x)=-1/(qx)$ has only one fixed point given by
$iq^{-1/2}$. This equals $[i]_{q}$, the $q$-deformed version of $i$ from [17,
formula (9)]. The translation $T_{q}(x)=qx+1$ has two fixed points at the
(usual) boundary at infinity, given by $\infty$ and $1/(1-q)$. However, we
expect the boundary of $\mathbb{H}^{2}_{q}$ to be
$\mathbb{R}_{q}\cup\\{\infty\\}$, where $\mathbb{R}_{q}$ denotes the
$q$-reals. On $\mathbb{R}_{q}$, the transformation $T_{q}$ has no fixed point
since $T_{q}[x]_{q}=[x+1]_{q}$.

An important role should play the $q$-rational transition map
$g_{q}(x)=\frac{1+(x-1)q}{1+(q-1)x}$. Since it deforms the identity, there are
strictly more transformations in the deformed setting. For ${q\neq 1}$,
$g_{q}$ is an elliptic transformation with only fixed point on
$\mathbb{H}^{2}$ given by $\frac{1+{\rm i}\sqrt{3}}{2}$ which is independent
of $q$. In [17, Part 2.3], it is shown that this complex number stays itself
under $q$-deformation. Note that both transformations $g_{q}$ and $T_{q}S_{q}$
are rotations around the same center. Hence they commute. Similarly, the
matrix of $g_{q}^{-1}$ anti-commutes with the matrix of $S_{q}$.

These links between $q$-deformed numbers and the $q$-deformed
$\mathfrak{sl}_{2}$-algebra are intriguing and might point towards a deeper
relation.

### Acknowledgements

I warmly thank Valentin Ovsienko and Sophie Morier-Genoud for inspiration,
many suggestions and fruitful exchanges, and Peter Smillie and Vladimir Fock
for helpful discussions. I also thank the anonymous referees for their remarks
improving the paper. I gratefully acknowledge support from the University of
Heidelberg where this work has been carried out, in particular under ERC-
Advanced Grant 101018839 and Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - Project-ID 281071066 - TRR 191.

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# The pyglaf argumentation reasoner (ICCMA2021)

Mario Alviano
Department of Mathematics and Computer Science
University of Calabria
Rende (CS), IT 87036
<EMAIL_ADDRESS>

###### Abstract

The pyglaf reasoner takes advantage of circumscription to solve computational
problems of abstract argumentation frameworks. In fact, many of these problems
are reduced to circumscription by means of linear encodings, and a few others
are solved by means of a sequence of calls to an oracle for circumscription.
Within pyglaf, Python is used to build the encodings and to control the
execution of the external circumscription solver, which extends the SAT solver
glucose and implements algorithms taking advantage of unsatisfiable core
analysis and incremental computation.

##  1 Introduction

Circumscription [1] is a nonmonotonic logic formalizing common sense reasoning
by means of a second order semantics, which essentially enforces to minimize
the extension of some predicates. With a little abuse on the definition of
circumscription, the minimization can be imposed on a set of literals, so that
a set of negative literals can be used to encode a maximization objective
function. Since many semantics of abstract argumentation frameworks are based
on a preference relation that essentially amount to inclusion relationships,
pyglaf (http://alviano.com/software/pyglaf/) uses circumscription as a target
language to solve computational problems of abstract argumentation frameworks.

pyglaf [2] is implemented in Python and uses circumscriptino
(http://alviano.com/software/circumscriptino/), a circumscription solver
extending the SAT solver glucose [3]. Linear reductions are used for all
semantics [4]. For the ideal extension, the reduction requires the union of
all admissible extensions of the input graph; such a set is computed by means
of iterative calls to circumscriptino. The communication between pyglaf and
circumscriptino is handled in the simplest possible way, that is, via stream
processing. This design choice is principally motivated by the fact that the
communication is often minimal, limited to a single invocation of the
circumscription solver.

The reasoner supports the following ICCMA 2021 problems: CE-CO, CE-PR, CE-SST,
CE-ST, DC-CO, DC-PR, DC-SST, DC-ST, DC-STG, DS-CO, DS-ID, DS-PR, DS-SST, DS-
ST, DS-STG, SE-CO, SE-ID, SE-PR, SE-SST, SE-ST, and SE-STG. Support for CE-STG
is a work in progress.

##  2 Circumscription

Let $\mathcal{A}$ be a fixed, countable set of _atoms_ including $\bot$. A
_literal_ is an atom possibly preceded by the connective $\neg$. For a literal
$\ell$, let $\overline{\ell}$ denote its _complementary literal_ , that is,
$\overline{p}=\neg p$ and $\overline{\neg p}=p$ for all $p\in\mathcal{A}$; for
a set $L$ of literals, let $\overline{L}$ be $\\{\overline{\ell}\mid\ell\in
L\\}$. _Formulas_ are defined as usual by combining atoms and the connectives
$\neg$, $\wedge$, $\vee$, $\rightarrow$, $\leftrightarrow$. A _theory_ is a
set $T$ of formulas including $\neg\bot$; the set of atoms occurring in $T$ is
denoted by $\mathit{atoms}(T)$. An _assignment_ is a set $A$ of literals such
that $A\cap\overline{A}=\emptyset$. An _interpretation_ for a theory $T$ is an
assignment $I$ such that
$(I\cup\overline{I})\cap\mathcal{A}=\mathit{atoms}(T)$. Relation $\models$ is
defined as usual. $I$ is a _model_ of a theory $T$ if $I\models T$. Let
$\mathit{models}(T)$ denote the set of models of $T$.

_Circumscription_ applies to a theory $T$ and a set $P$ of literals subject to
minimization. Formally, relation $\leq^{P}$ is defined as follows: for $I,J$
interpretations of $T$, $I\leq^{P}J$ if $I\cap P\subseteq J\cap P$.
$I\in\mathit{models}(T)$ is a _preferred model_ of $T$ with respect to
$\leq^{P}$ if there is no $J\in\mathit{models}(T)$ such that $I\not\leq^{P}J$
and $J\leq^{P}I$. Let $\mathit{CIRC}(T,P)$ denote the set of preferred models
of $T$ with respect to $\leq^{P}$.

##  3 From Argumentation Frameworks to Circumscription

An _abstract argumentation framework_ (AF) is a directed graph $G$ whose nodes
$\mathit{arg}(G)$ are arguments, and whose arcs $\mathit{att}(G)$ represent an
attack relation. An _extension_ $E$ is a set of arguments. The _range_ of $E$
in $G$ is $E^{+}_{G}:=E\cup\\{x\mid\exists yx\in\mathit{att}(G)$ with $y\in
E\\}$. In the following, the semantics of ICCMA’17 are characterized by means
of circumscription.

For each argument $x$, an atom $a_{x}$ is possibly introduced to represent
that $x$ is attacked by some argument that belongs to the computed extension
$E$, and an atom $r_{x}$ is possibly introduced to enforce that $x$ belongs to
the range $E^{+}_{G}$:

$\displaystyle\mathit{attacked(G)}:={}$
$\displaystyle\left\\{a_{x}\leftrightarrow\bigvee_{yx\in\mathit{att}(G)}{y}\
\middle|\ x\in\mathit{arg}(G)\right\\}$ (1)
$\displaystyle\mathit{range(G)}:={}$ $\displaystyle\left\\{r_{x}\rightarrow
x\vee\bigvee_{yx\in\mathit{att}(G)}{y}\ \middle|\
x\in\mathit{arg}(G)\right\\}$ (2)

The following set of formulas characterize semantics not based on preferences:

$\displaystyle\mathit{conflict\textit{-}free}(G):={}$
$\displaystyle\\{\neg\bot\\}\cup\\{\neg x\vee\neg y\mid
xy\in\mathit{att}(G)\\}$ (3) $\displaystyle\mathit{admissible}(G):={}$
$\displaystyle\mathit{conflict\textit{-}free}(G)\\!\cup\\!\mathit{attacked}(G)\cup\\{x\rightarrow
a_{y}\mid yx\in\mathit{att}(G)\\}$ (4) $\displaystyle\mathit{complete}(G):={}$
$\displaystyle\mathit{admissible}(G)\cup\left\\{\left(\bigwedge_{yx\in\mathit{att}(G)}{a_{y}}\right)\rightarrow
x\ \middle|\ x\in\mathit{arg}(G)\right\\}$ (5)
$\displaystyle\mathit{stable}(G):={}$
$\displaystyle\mathit{complete}(G)\cup\mathit{range}(G)\cup\\{r_{x}\mid
x\in\mathit{arg}(G)\\}$ (6)

Note that in (4) truth of an argument $x$ implies that all arguments attacking
$x$ are actually attacked by some true argument. In (5), instead, whenever all
attackers of an argument $x$ are attacked by some true argument, argument $x$
is forced to be true. Finally, in (6) all atoms of the form $r_{x}$ are forced
to be true, so that the range of the computed extension has to cover all
arguments.

The ideal semantic is defined as follows (Proposition 3.6 by [5]): Let $X$ be
the set of admissible extensions of $G$ that are not attacked by any
admissible extensions, that is,
$X:=\\{E\in\mathit{models}(\mathit{admissible}(G))\mid\nexists
E^{\prime}\in\mathit{models}(\mathit{admissible}(G))$ such that
$yx\in\mathit{att}(G),x\in E,y\in E^{\prime}\\}$. $E$ is the ideal extension
of $G$ if $E\in X$, and there is no $E^{\prime}\in X$ such that
$E^{\prime}\supseteq E$.

All semantics of ICCMA’19 are characterized in circumscription as follows:

$\displaystyle\mathit{co}(G):={}$
$\displaystyle\mathit{CIRC}(\mathit{complete}(G),\emptyset)$ (7)
$\displaystyle\mathit{st}(G):={}$
$\displaystyle\mathit{CIRC}(\mathit{stable}(G),\emptyset)$ (8)
$\displaystyle\mathit{gr}(G):={}$
$\displaystyle\mathit{CIRC}(\mathit{complete}(G),\mathit{arg}(G))$ (9)
$\displaystyle\mathit{pr}(G):={}$
$\displaystyle\mathit{CIRC}(\mathit{complete}(G),\overline{\mathit{arg}(G)})$
(10) $\displaystyle\mathit{sst}(G):={}$
$\displaystyle\mathit{CIRC}(\mathit{complete}(G)\cup\mathit{range}(G),\\{\neg
r_{x}\mid x\in\mathit{arg}(G)\\})$ (11) $\displaystyle\mathit{stg}(G):={}$
$\displaystyle\mathit{CIRC}(\mathit{conflict\textit{-}free}(G)\cup\mathit{range}(G),\\{\neg
r_{x}\mid x\in\mathit{arg}(G)\\})$ (12) $\displaystyle\mathit{id}(G,U):={}$
$\displaystyle\mathit{CIRC}(\mathit{admissible}(G)\cup\overline{\mathit{arg}(G)\setminus
Y},\ \overline{Y})$ (13)

where in (13) $U$ is the union of all admissible extensions of $G$, and $Y$ is
$U\setminus\\{x\mid\exists yx\in\mathit{att}(G),\ y\in U\\}$.

##  4 Implementation

Abstract argumentation frameworks can be encoded in trivial graph format (tgf)
as well as in aspartix format (apx). The following data structures are
populated during the parsing of the input graph $G$: a list arg of the
arguments in $\mathit{arg}(G)$; a dictionary argToIdx, mapping each argument
$x$ to its position in arg; a dictionary att, mapping each argument $x$ to the
set $\\{y\mid xy\in\mathit{att}(G)\\}$; a dictionary attR, mapping each
argument $x$ to the set $\\{y\mid yx\in\mathit{att}(G)\\}$. Within these data
structures, theories (7)–(13) are constructed in amortized linear time. Single
extension computation and extension enumeration is then demanded to the
underlying circumscription solver [6].

The union $U$ of all admissible extensions is computed by iteratively asking
to circumscriptino to compute an admissible extension that maximize the
accepted arguments not already in $U$, so to expand $U$ as much as possible at
each iteration.

For complete, stable, and preferred extensions, credulous acceptance is
addressed by checking consistency of the theory extended with the query
argument. Similarly, skeptical acceptance is addressed by adding the
complement of the query argument for complete, and stable extensions. Grounded
and ideal extensions are unique, and therefore credulous acceptance is
addressed by checking the presence of the query argument in the computed
extension. Actually, for the ideal extension, a negative answer is possibly
produced already if the query argument is not part of the union of all
admissible extensions. The remaining acceptance problems are addressed by a
recent algorithm for query answering in circumscription [7]. In a nutshell,
given a query atom $q$ and a circuscribed theory, the computational problem
amounts to search for a model of the circumscribed theory that contains the
query atom. The algorithm implemented in circumscriptino searches for a
classical model of the theory that contains the query atom, and checks that no
more preferred model not containing the query atom exists. In this way,
queries are possibly answered without computing any optimal model.

Counting of extensions is implemented by enumeration, with no particular
optimization.

## Acknowledgments

This work was partially supported by the projects PON-MISE MAP4ID (CUP
B21B19000650008) and PON-MISE S2BDW (CUP B28I17000250008), by the LAIA lab
(part of the SILA labs) and by GNCS-INdAM.

## References

  * [1] John McCarthy.  Circumscription - A form of non-monotonic reasoning.  Artif. Intell., 13(1-2):27–39, 1980.
  * [2] Mario Alviano.  The pyglaf argumentation reasoner.  In Ricardo Rocha, Tran Cao Son, Christopher Mears, and Neda Saeedloei, editors, TC of ICLP 2017, volume 58 of OASICS, pages 2:1–2:3. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017.
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  * [7] Mario Alviano.  Query answering in propositional circumscription.  In Jérôme Lang, editor, Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, IJCAI 2018, July 13-19, 2018, Stockholm, Sweden., pages 1669–1675. ijcai.org, 2018.