Infinitesimal Modular Group: $q$-Deformed $\mathfrak{sl}_2$ and Witt Algebra

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 M\"obius transformations acting on the hyperbolic plane.


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 PSL 2 (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 = id = (ST ) 3 .
The q-deformed integers [n] q = 1 + q + q 2 + • • • + q n−1 = 1−q n 1−q , where q ∈ 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 2 q = id = (S q T q ) 3 .The qrational 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 = q 1 0 1 and S q = 0 −1 q 0 .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 σ 1 = T q and σ 2 = S q T q S q = 1 0 −q q .One easily checks the braid relation σ 1 σ 2 σ 1 = σ 2 σ 1 σ 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 qreal numbers?Basic functions on real numbers are monomials and the exponential function, which are eigenfunctions of the vector fields associated to sl 2 (R) acting on the completed line RP 1 .The goal of our investigation is to q-deform these vector fields and to analyze their eigenfunctions.
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 PSL 2 (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: where S acts on the space of functions by precomposition.We introduce the differential operator D 0 , a q-deformed version of x∂, given by We will see that D 0 anti-commutes with S q .Together with D 1 := S q •D −1 •S q , a deformation of x 2 ∂, 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 The theorem tells us that the module over R[q] generated by D −1 , D 0 and D 1 is a deformation of the Lie algebra sl 2 (R) which we recover for q = 1.The Lie algebra sl 2 being simple, it does not allow for non-trivial deformations.Hence our deformation is isomorphic to sl 2 as a Lie algebra, but they are different as Z[q]-modules.This is similar to quantum groups.
A fundamental role is played by the Möbius transformation 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 This theorem comes from the interplay between g q , T q and S q given by g q • T q = T q −1 • g q and g q • S q = S q −1 • 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 −1 q = 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) D 0 (g q ) = q 2 − q + 1 g q , D −1 (g q ) = q + (1 − q)g q , D 1 (g q ) = (q − 1)g q + g 2 q , (2) 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 (ℓ n ) n∈Z with bracket given by This algebra can be realized as differential operators (or equivalently as vector fields) via ℓ n = x n−1 ∂.Putting for n > 1: we get a deformation of the Witt algebra (see Theorem 3.2): Theorem 1.4.The (D n ) n∈Z form a Lie algebra with bracket given by (where n, r > 0): 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 qdeformed sl 2 and the quantum group U q (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 C x ±1 and using the q-differential ∂ q (f ) = f (qx)−f (x) qx−x .This was generalized in [8] to more general σ-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 is studied.Yet another proposal from [16, formula (3.18)] gives operators D n (q) for n ∈ Z with commutator [D n (q), D m (q)] = q − q −1 [n − m] q D n+m q 2 (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 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.

Deformed sl and Heisenberg algebra
The group SL 2 acts naturally on the projective line P 1 .We will work over R or C. Differentiating this action at the identity gives a realization of the Lie algebra sl 2 as vector fields on P 1 .Using the two standard charts of P 1 with transition function x → 1/x, the image of sl 2 → Vect P 1 is generated by ∂, x∂ and x 2 ∂ written in the first chart, where we use the notation ∂ = d/dx.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 sl 2 .Together with a q-deformed identity map, we deform the 3-dimensional Heisenberg algebra.

Deformed sl 2
On P 1 , consider the Möbius transformations where q ∈ C * is fixed or seen as a formal parameter.They deform the translation x → x + 1 and the inversion x → −1/x.These transformations act on the space of functions on P 1 by precomposition.
Consider the differential operator D −1 on P 1 which is defined in the first chart by 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 On the other side, 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. 1 To find E q , one has to solve x .The solution is given by 1 Unpublished, private communication.
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: 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 Proof .For a function f (x), we have on the one side On the other hand, More generally, we can find all operators D of the form p(x)∂ which anti-commute with S q .The relation {D, S q } = 0 gives 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∂ 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 α.One has to solve αf = D 0 f , i.e., (ln f We define the q-rational transition map 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)∂ 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 ∈ Z[q].We impose that D deforms x∂, that is p 0 (1) = 0 and p 1 (1) = 1.We also impose the leading terms of p 0 , p 1 to be ±1.We wish that the eigenfunctions of D are A. Thomas Möbius transformations in Z[q].This is only the case if the discriminant of p 0 + p 1 x − qp 0 x 2 is a square in Z[q].This leads to the equation p 1 (q) 2 + 4qp 0 (q) 2 = R(q) 2 for some R ∈ Z[q].This is equivalent to 4qp 2 0 = (R − p 1 )(R + p 1 ).Excluding the case where p 0 = 0 which leads to the undeformed operator x∂, 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 sl 2 .For that, we wish to deform x 2 ∂.Note that x 2 ∂ = S • ∂ • S.This motivates the following definition: 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 sl 2 : Theorem 2.3.The operators D −1 , D 0 and D 1 form a Lie algebra with brackets For q = 1, we get the Lie algebra sl 2 .
Proof .The proof is a straightforward computation.All D i are of the form g(x)∂ 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 Z Similarly, we have The last bracket can be computed to be [D −1 , D 1 ] = 1 − q + 2qx + q(q − 1)x 2 ∂.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 sl 2 being simple, it does not allow any non-trivial deformations.Our qdeformation is indeed abstractly isomorphic to sl 2 when q and q 2 − q + 1 are invertible.To give an explicit isomorphism, denote by (f, h, e) the generators of sl 2 given by the differential operators ∂, x∂, x 2 ∂ .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): Using this isomorphism to 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 = ( 0 0 1 0 and e = 0 −1 0 0 , we get Note that this representation is not in sl 2 (Q[q]).A direct computation shows that there is no 2-dimensional representation of our q-deformed sl 2 into sl 2 (Q[q]).In dimension 3, there is of course the adjoint representation into sl 3 (Z[q]).
The main drawback of this choice is that the eigenfunctions of D 0 are Möbius transformations with coefficients not in Z[q].Remark 2.5.A simpler and very similar Lie algebra deforming sl 2 is given by generators 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.

q-rational transition map
The map g q defined in (2.1) plays a fundamental role, both for generalizing the q-deformation from 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 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 which is of determinant q 2 − q + 1.For q ̸ = 1, g q is an elliptic transformation since its normalized trace is given by q + 1 The unique fixed point on 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 × 2 matrices satisfy: g q T q = qT q −1 g q and g q S q = qS q −1 g q .
Therefore, seen as Möbius transformations, we have g q • T q = T q −1 • g q and g q • S q = S q −1 • g q .
As an application, we can reprove the positivity property of left q-rationals, proven in [2, Appendix A.1] via an explicit combinatorial interpretation.
Proof .From [12, Proposition 1.3], we know that for r/s ∈ Q >1 the right q-rational [r/s] ♯ q = R ♯ (q)/S ♯ (q) is a rational function in q with positive coefficients, i.e., R ♯ , S ♯ ∈ N[q].We also know from [12,Theorem 2] . Since r/s+1 > r/s and [r/s+1] ♯ q = q[r/s] ♯ q +1, we also get (q − 1)R ♯ + S ♯ ∈ N[q].Finally, we deduce that r s has positive coefficients.Multiplying both numerator and denominator with an appropriate power of q, we get the same for [r/s] ♭ q .■ Finally, we can use the transition function g q (x) as reparametrization of P 1 .To emphasize the dependence of our differential operators, we will write here D −1 (q, x) = (1 + (q − 1)x)∂ x and similar for D 0 and D 1 .
Proposition 2.9.Reparametrizing P 1 by the transition map ξ = g q (x) gives The behavior of D −1 and D 1 is reminiscent of Proposition 2.6.
Proof .Using dξ dx = q 2 −q+1 (1+(q−1)x) 2 and x = ξ+q−1 q+(1−q)ξ , we get 1 + (q − 1)x = q 2 −q+1 q+(1−q)ξ .Hence The computation for D 1 is similar.Finally, This proposition indicates that we can use the undeformed operator x∂ together with D ±1 to get a deformation of 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∂.

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.
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: 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 : 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 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: We can use this proposition to express one operator in terms of another via the relation D i = D i (gq) D j (gq) D j for all i, j ∈ {−1, 0, 1}.This holds true since these differential operators are of order 1.

Deformed Witt algebra
Now that we have deformed the differential operators ∂, x∂ and x 2 ∂, we can do the same for all x n ∂ for n ∈ 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 ℓ n = x n+1 ∂, the Lie algebra structure is given by To get a deformation of the Witt algebra, we define for n > 1: where g −1 q = 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 ≥ 1 and g q D −n = qD −n+1 for n ≥ 2. In addition, Similarly, there is a nice behavior when multiplied by qg −1 q .By definition qg −1 q D −n = D −n−1 for n ≥ 1 and qg −1 q D n = qD n−1 for n ≥ 2. In addition, Proof .From the definitions, we get g q D 0 = (1 + (x − 1)q) 2 ∂.From Proposition 2.3 and its proof, we see that this is [D For the second half, note that Dividing by g q gives qg −1 q D 1 = D 0 + (q − 1)D −1 .Similarly, (q − 1)g q D 0 + q 2 − q + 1 D −1 = qD 0 , so dividing by g q gives qg −1 q D 0 = (q − 1)D 0 + q 2 − q + 1 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∈Z form a Lie algebra with bracket given by (with n, r > 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, 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: 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 ≥ 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) An easy induction gives for r > 0, Also we get g r q D −1 = g r−1 q D 0 + (1 − q)D r , where we can use equation (3.2) to express the first term.Similar results hold for qg −1 q r D 0 and qg −1 ■ Remark 3.3.Regarding Remark 2.5, we could try to simplify the defining relations of the qdeformed 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 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.

Möbius transformations
The differential operators ∂, x∂, x 2 ∂ can be interpreted in at least three different ways: first as differential operators on P 1 written in one chart (this was our approach).Second they can be seen as complex vector fields on the circle S 1 ⊂ 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 sl 2 this symmetric space is the hyperbolic plane H 2 .In this section, we integrate the operators D −1 , D 0 and D 1 seen as vector fields of H 2 .The result gives interesting Möbius transformations with q-parameter.In the Taylor expansion around q → 1 (the "semi-classical limit"), we recover the deformed translation T q .Conjecturally there should be a q-deformation of H 2 on which these transformations act, such that the boundary can be identified with the q-deformed real numbers of [13].

Classical setting
Consider first the classical setup with the operators ∂, x∂ and x 2 ∂.These are Killing vector fields on the hyperbolic plane H 2 , whose integration determines isometries of H 2 .Here, we consider H 2 ⊂ C in the upper half-plane model and use the coordinate x ∈ C.
To start, consider the case of the vector field V = ∂ = ∂ ∂x .A curve γ integrates this vector field iff γ ′ (t) = V (γ(t)) = 1 for all t ∈ R (where we identified 1 with the constant vector field ∂).With initial condition γ(0) = x we get γ(t) = x + t.We should think of this as a function γ x (t) of the initial condition.For time 1, we get the translation γ x (1) = x + 1 = T (x).
Another important case is x∂, for which we have to solve γ ′ (t) = γ(t).With initial condition γ(0) = x, we obviously get γ(t) = e t x.The function x → e t x is the hyperbolic isometry of H 2 associated to the geodesic joining 0 to ∞.Its matrix is given by e t/2 0 0 e −t/2 . (4.1) We can immediately generalize to the generators of the Witt algebra.Consider the operator x n ∂ with n ∈ Z, n ̸ = 1.A curve γ integrates the associated vector field if γ ′ (t) = γ(t) n .The solution with initial condition γ(0) = x is given by 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 γ x (t) n−1 = y 1 − (n − 1)ty .

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)∂ 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)∂.The associated differential equation is γ ′ (t) = 1 + (q − 1)γ(t) with initial condition γ(0) = x.Solving this equation is standard: first one solves the homogeneous equation, then one uses the variation of the constant to finally get For t = 1, we get the associated map x → − 1 q−1 + x + 1 q−1 e q−1 .The Taylor expansion around q − 1 at order 1 gives which is nothing but T q (x).For a general time t, the same procedure gives x → (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 H 2 is the affine map x → − 1 q−1 + x + 1 q−1 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 .
We already see that γ x (t) is a Möbius transformation in x since C 2 is.For time t = 1, we get the following.
If we Taylor expand γ 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 × 2 matrix to order 1 in q − 1.The result is W q := e(1 − 2q) (e − 1)(q − 1) (e − 1)(q − 1) −q , where we used q 2 = 2q − 1 coming from the Taylor expansion.We see that q = 1 gives the transformation x → ex.A similar computation with arbitrary time t gives W t q = e t (t − qt − q) (e t − 1)(q − 1) (e t − 1)(q − 1) −q , which for q = 1 gives the transformation x → e t x from equation (4.1).

Speculations about a q-deformed hyperbolic plane
The above computations seem to indicate the existence of a q-deformed version of the hyperbolic plane 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 ∞ and 1/(1 − q).However, we expect the boundary of H 2 q to be R q ∪ {∞}, where R q denotes the q-reals.On 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) = 1+(x−1)q 1+(q−1)x .Since it deforms the identity, there are strictly more transformations in the deformed setting.For q ̸ = 1, g q is an elliptic transformation with only fixed point on H 2 given by 1+i √ 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 −1 q anti-commutes with the matrix of S q .These links between q-deformed numbers and the q-deformed sl 2 -algebra are intriguing and might point towards a deeper relation.

Remark 2 . 4 .
It is tempting to consider D −1 , D 1 and D 0 := [D −1 , D 1 ].The operator D 0 still anti-commutes with S q and the bracket relations are

q 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 ].