Hom-Lie-Virasoro symmetries in Bloch electron systems and quantum plane in tight binding models

We discuss the Curtright-Zachos (CZ) deformation of the Virasoro algebra and its extentions in terms of magnetic translation (MT) group in a discrete Bloch electron system, so-called the tight binding model (TBM), as well as in its continuous system. We verify that the CZ generators are essentially composed of a speciﬁc combination of MT operators representing deformed and undeformed U (1) translational groups, which determine phase factors for a ∗ -bracket commutator. The phase factors can be formulated as a ∗ - ordered product of the commutable U (1) operators by interpreting the AB phase factor of discrete MT action as ﬂuctuation parameter q of a quantum plane. We also show that some sequences of TBM Hamiltonians are described by the CZ generators


FFZ and Moyal deformation
The Moyal sine algebra (FFZ algebra) [1] is related to the Moyal bracket deformation, which is well-known as a Lie-algebraic deformation of the Poisson brackets and has been applied to describe noncommutative phenomena in various regions such as noncommutative geometry in string theories and quantum Hall physics.
(1. 5) In the context of deformation quantization one may keep the star product structure [τ n,k , τ m,l ] * = 2i sin( θ 2 (nl − mk))τ n+m,k+l . (1.6) What we call the FFZ algebra in this note is originally given by (1.5) with the introduction of deformation parameter q and the q-bracket for an arbitrary object A q = e i θ , and Changing the normalization T (n,k) = 1 q − q −1 τ n,k (1.8) we have the FFZ algebra where we assume that τ n,k is generalized to an arbitrary operator behaving like the Moyal star products τ n,k τ m,l = q nl−mk 2 τ n+m,k+l . (1.10) It is well known that magnetic translation (MT) operators satisfy this fusion relation [3].

noncommutative geometry
Quantum field theory on noncommutative spaces [4] is one of many applications of the Moyal brackets.
The coordinates of the endpoints of open strings constrained to a D-brane in the presence of a constant Neveu-Schwarz B field are reported to be relevant to a noncommutative algebra [5]. The field theory on noncommutative boundary space (known as noncommutative field theory) has attracted much attention in string and M-theories, and the noncommutativity is expressed in the Moyal star bracket [x µ , x ν ] * = i θ µν .
This noncommutative feature is understood to be originated from the same idea as noncommutative magnetic translations in two-dimensional quantum mechanics in a constant magnetic field [3].
There is also an interesting topic in quantum gravity in a relevance to infinite dimensional symmetries.

quantum Hall effect
On the other hand, another type of W∞ algebra has been examined in quantum Hall physics [9,10] and in the conformal field theory of edge excitations as well as in bulk physics extension [11]. It is also known that W∞ is associated to the area-preserving diffeomorohisms (for a review see [12]) of incompressible fluids.

quantm algebra and Hall effect
Another interesting aspect of quantum Hall effect is a certain relevance to integrable models in twodimensional lattice systems. The algebraic approach of dynamical symmetries is useful when the Hamiltonian can be written in terms of the symmetry generators. The problem of Bloch electrons in a constant magnetic field can be solved by making use of a relation between the group of magnetic translations and Uq(sl2) quantum group [15] (strictly speaking the quantum algebra, which is a universal enveloping algebra with non-cocommutative Hopf algebra structure [15,16,17,18]). The Hamiltonian of a particle on a twodimensional square lattice in the magnetic field (tight binding model) is composed of Uq(sl2) raising and lowering operators, which are expressed in N -dimensional Weyl bases X and Y with the commutation relation qXY = Y X [19]. The same approach on a triangluar lattice is shown in [20] adding the 3rd basis Z satisfying qZX = XZ and qY Z = ZY . Spectra of these models can be represented by means of solutions of Bethe ansatz type algebraic equations [19,20,21]. The magnetic translations in the discretized systems are given by products of the Weyl base matrices, and their commutation relations are accompanied by a global phase factor such as q 2 TxTy = TyTx owing to the Weyl commutation relations.
While in a continuous coordinate system, magnetic translations are made of differential operators, and their phase factors are no longer global but rather comprising local parameters as seen in (1.6), and the Hamiltonian cannot be written in terms of those quantum group generators. Nevertheless it is interesting that there exist the generators of quantum algebra Uq(sl2) expressed by magnetic translations [22] in somewhat parallel form to the cases of the lattice systems. Furnished with the local parameters of Moyal type (1.6), the generators can further be extended to q-deformed Virasoro (super)algebras [23,24] which are studied in field theory context [25,26,27,28,29,30]. Although these deformed algebras are not a quantum algebra but infinite dimensional Lie algebras, an interesting point is definitely an appearance of Moyal noncommutative feature of magnetic translations. Despite the clear correspondence of the tight binding Hamiltonian to the continuous system in a continuum limit, the inheritance of properties of magnetic translations such as quantum groups and the Weyl bases is still unclear. In other words, they are considered to disappear or not to be in the limit, regardless of having the similar characteristics mentioned above.

quantum space and CZ algebra
In order to reveal the truth and resolve such a conflicting situation, quantum groups and planes may help.
Quantum groups [31] are another notion to describe noncommutative geometry, and they are formulated to be dual objects to quantum algebras [32,33]. They are deformations of matrix groups whose matrix entries obey certain commutation relations depending on a deformation parameter q. Quantum space possesses noncommuting coordinates X i and their differentials ∂i = ∂ ∂X i which satisfy the relations where the coefficient matrices B, C and F satisfy certain Yang-Baxter relations [34]. This ensures that the coordinates and their derivatives behave covariantly under the action of quantum group matrix. One of intriguing and the simplest example is the bosonic part of GLq(1, 1) quantum superspace since the Virasoro operators in this quantum space satisfy another version of q-deformed Virasoro algebra called the Curtright-Zachos (CZ) algebra [35] [ There are many results concerning representations of the CZ algebra such as: q-harmonic oscillators [36], central extensions [37,38] and operator product formula (OPE) [38], matrix representation [39], quantum space diffrential calculi [40,41,42], and fractional spin representation [43]. Multi-parameter deformations [44,45] and supersymmetric extensions [40,41,46] are also studied. Motivated by its application to physical systems, several investigations have been made: deformation of soliton equations [47,48], transformation to commutator form [48], Jacobi consistency conditions [38,49], and so on.

Contents
The purpose of this paper is three fold: first one is to clarify a dynamical origin of the phase factors used in the commutator deformation of the algebra (1.19). Second is to clarify a relation between the phase factors and quantum plane picture which is relevant to physical models that possess a quantum algebra symmetry.
Third is to present various ways of constructing the CZ generators. We investigate some properties of the CZ generators based on the algebras of MT and discrete magnetic translation (DMT), and discuss their relations to the Hamiltonian systems relevant to a tight binding model (TBM), which is a discrete model of a particle on a two-dimensional lattice in constant magnetic field. We expect to catch a glimpse of quantization of space, or rather the quantum plane structure, as a natural effect of space discretization in TBM, in addition to which is known to possess the quantum algebra symmetry Uq(sl2) (see Appendix A).
In Section 2, reviewing some properties of CZ algebra, we explain mathematical settings and MT algebras as our basic tool. The magnetic translations in TBM are reviewed in Section 5. In Section 3, we consider MT realizations of CZ algebras starting from a q-derivative representation of the algebra. We present a generalized algebra CZ * , which includes CZ algebra as a subalgebra of it. The generators of these algebras are composed of certain combinations of specific MT operators, and all these algebras as well as MT algebra can be encapsulated into a single expression of deformed commutators with a * -product structure. 1) The structure of CZ * generators consists of commutative MT and noncommutative MT parts. The commutative part plays the role of the fundamental CZ algebraic relation (structure constant of the algebra), and the noncommutative part plays the role of a nonlocal operator of * -commutative translation (deformed U (1)) and the role of defining a weight for * -product phase factors as well.
In Section 4, we construct a matrix representation of CZ * and verify the property of the representation given in Section 3 (we call the representation in Section 3 "commutative representation"). we investigate a mechanism how quantum plane structure and the * -bracket structure arise in CZ * algebra. CZ * contains two subalgebras CZ ± , and they correspond to two orthogonal directions on quantum plane in TBM. Section 4.2 outlines the role and significance of commutative representations. Section 4.3 defines the algebra family of CZ * representations that provide the TBM Hamiltonian sequences, in preliminary for the specific verification explained in Section 5. Section 4.4 presents another definition of our * -product. We first show that the commutative representations are composed of composite operators of DMT units in two directions. Then 1) It is not clear that our * -product is the same as the Moyal * -product, however for convenience of terminology we adopt the same symbol without causing any serious confusion making use of the commutative representations and introducing a concept of * -ordering product, we formulate the appearance of the phase factor of the CZ * commutators on basis of the quantum plane picture of DMT in TBM. The derivation of the matrix representations of DMT is shown in Appendix B, where confirmation of their MT algebra is made as well.
In Section 5, we discuss the TBM Hamiltonian series universally represented by the matrix representations of CZ * algebras. In Section 5.1, we derive the DMT algebra (exchange, fusion and circulation rules) and observe the correspondence between the AB phase and the quantum plane fluctuations accompanied by the DMT movement. In Section 5.2, we describe the TBM Hamiltonian sequenceȞ k using the n = ±1 modes of CZ * family generators. Section 5.3 discusses extensions to general modes of CZ * family in accord to the Hamiltonian systemsĤn with the effective spacing of magnetic lattice extended from 1 to n. In Section 5.4, more generic sequenceĤ (n,k) with the quantum plane fluctuation (power of q) extended from 1 to k is represented by the genuine CZ * generators. As an aid to understanding some formulae used in Section 5, we review the quantum group symmetry of the original TBM Hamiltonian, and add some remarks on q-inversion symmetry of the Hamiltonian in Appendix A.

Curtright-Zachos (CZ) algebra
We make a brief review of mathematical settings on the CZ algebra. There are two types of bracket deformation, and they differ in how they introduce their phase factors. The way of introducing the phase factors depends on explicit realizations of the generators of CZ algebra. Namely, there is no a priori way to determine how to introduce the phase factors and we have to start with a realization of the generators.
However, the only known realization with physical implications is the deformation of harmonic oscillators, and we hence need other realizations to explore physical and mathematical properties of the CZ algebra.
To this end, in Section 2.2, we introduce magnetic translation (MT) algebra, which is a typical realization of the Moyal sine algebra (FFZ algebra) in physical system. The property of generating phase factors when MT operators commute is compatible with the deformed commutation relation of the CZ algebra, and it is very convenient to investigate various features of the CZ algebra. Using MT representations in a later section, we will explain that there exists a larger algebra containing the CZ algebra, and investigate * -bracket formulation for these algebras altogether.

bracket deformation
The CZ algebra is neither a standard Lie algebra nor a quantum algebra, but a Hom-Lie algebra [50,51,52], which is a deformed Lie algebra satisfying skew symmetry and Hom-Jacobi conditions [52]. Representations in q-harmonic oscillator and quantum superspace as well as (1.18) are known to satisfy these conditions [40,41,48].
Another feature that distinguishes it from the usual Virasoro algebra is the existence of the central satisfying the commutation relations Unlike the standard quantum and Lie algebras, the CZ algebra is defined in terms of the deformed bracket. We have two options to express the CZ algebra (1.19) according to the way of defining deformed commutators: one is the exteria type its essential origin is, we suppose that and we then have In order to search for a certain origin of (2.10), we later investigate MT representations of CZ algebras in Section 3. In Section 4.4, we further explore mathematical features (quantum plane and * -ordered product) of discrete magnetic translations (DMTs) in tight binding model (TBM) on a two-dimensional square lattice as well.
2) We introduced the product as (AnBm)q = q m−n AnBm in the original paper [38].

magnetic translation algebra (MTA)
Magnetic translationsτR of a charged particle in a constant magnetic field on a cotinuous coordinate surface are written in terms of differential operators in the following nonlocal way (choice of unit φ0 = hc/e = 1)τ where TR is a translation by R, and ξ is a gauge fuction defined by (2.14) They satisfy the following algebraic relations (exchange and fusion rules): from which the circulation algebra is derived [3] τ −1 where φ is a magnetic flux (proportional to the area |R1 × R2| enclosed by the end points of circular operations). It is given by the differences of gauge function between two points where we denote R1 = (n1, m1) = (n, k) , R2 = (n2, m2) = (m, l) . (2.19) In this paper we refer the set of exchange, fusion and circulation rules as MTA (magnetic translation algebra).
Introducing the q parameter with magnetic length lB, unit length a and magnetic field B, under the choice of symmetric gauge we havê Applying the change of normalization to (2.22) we have the realization (1.9) of FFZ algebra [1] by means of MTA As a simple example ofτ (k) n let us have a look at angular momentum phase space (together with a spin value ∆) on a unit circle z = e iθ of cylinder coordinate w = ln z. Instead of the usual formτR = e i R·π described by the gauge covariant derivativesπi =pi + e c Ai, in this case we havê

* -bracket formulation and CZ * algebra
Starting from a q-differential operator expression of CZ generators, we present the realization of CZ algebra by MT operators, which leads to the idea of * -bracket formulation. In this section we discuss three types of CZ algebraic system CZ ± and CZ * . In Section 3.2, we show another special combination CZ − , which transforms into CZ + mutually with q-inversion. Both CZ ± can be combined into one algebraic system CZ * in the framework of * -brackets.

q-derivative representation of CZ algebra
Let us consider the q-analogue of differential operators ∂q (called the q-derivatives) defined by This satisfies the following Leibniz rule and the formula Defining the analogue of ln = −z n+1 ∂z by replacing ∂z with ∂q, we can verify thatLn satisfy the CZ algebra eq.(1.19). [ Noticing eqs.(3.1) and (3.4), the q-derivative can be rewritten by nonlocal expression of ordinary derivative ∂z, and we haveL This is related to the q-harmonic oscillators which satisfy the following relations Now, applying the magnetic translation (2.26) to (3.6) with the notation (2.24) we have another representation of the CZ generatorsL To make a contact with * -bracket formulation, we consider deformed commutators ofLn andT which are verified easily by utilizig the exchange relation between ∂q and q −kz∂ At this stage, there is no guideline for choosing one out of them. We hence need a limiting condition or a principle to constrain the form of phase factors to be attached to the commutator deformations.
One possible strategy is to follow the fact that magnetic translations should reduce to the usual translations when magnetic field vanishes. This naturally leads to the idea thatT Let us refer the upper index k inT (k) n to "weight" and define the weight ofLn to be 2. Considering the following set of operators for all integers n and k and allowing only the same phase factors as (3.16) forLn as well asT n , we thereby find the only three types of deformed commutators allowed where we have put k = 2 in the 2nd bracket reducing to (3.13), and k = l = 2 in the 3rd one reducing to (3.5). Note that we have omitted one type due to the skewness of the bracket (2.8) In this way we adopt eqs. In summary we confirm that * -bracket (2.9) can be defined for every element X and we organize eqs.(3.5),(3.13),(3.16) into the single common bracket form Note that the algebra of central element (2.6) reads if we regard the weight ofŜ k 0 as 2k in (3.20).

CZ ± and CZ * algebra
Inspecting the constitution of the CZ operatorsLn in terms of subalgebras ofT In order to examine the structure of CZ algebra (3.22), let us set the same phase factor forT (k) n as for Ln. Then we have If we put k, l = 0, 2 in this equation, we obtain the following closed subalgebras onT (k) n with weight 0 and 2, which are the parts ofLn defined in (3.11) According to these algebras, [Ln ,Lm ] (m−n) is verified to be closed without participations of any other weights: We should note thatT (0) n is a commuting local operator which does not include a differential operator w.r.t.

z, whileT
(2) n is a noncommuting nonlocal differential operator. However this noncommutativity feature is reversed each other in the view of deformed commutator world as can be seen in eq.(3.26). If we find a set of subalgebras similar to eqs. (3.26) and (3.27), it is possible to find another set of operators similar toLn.
It is in fact easy to find such a combination ofT (k) n by considering q-inverted version of (3.25) and setting k, l = 0, −2 we find another subalgebras corresponding to eqs. (3.26) and (3.27) [ Defining the q-inverted version ofLn with the new notationL ± where we denote the two algebras in (3.32) as CZ ± respectively. We notice that the signs of phase factors forL ± n in (3.32) are completely opposite, while the phase factors in (3.33) are not symmetric w.r.t. the exchange ofL ± n . This fact is originated in the q-inversion symmetrŷ However, this clumsy combination perfectly disappears if we incorporateL − n into M assuming the weight ofL − n to be −2. Thus changing its notation from M to we can make the new elementsL − n participate in the following extended algebras including (3.21)-(3.23) with (3.20): Although the two algebras CZ ± are expressed in (3.36) respectively, the remaining intersecting algebra * is yet open to incorporate into possible M * algebra. Using (3.31) We then find and its phase factors are verified to certainly be given by (3.20) with the weights k = ±2 applied to Now let us show that (3.36) and (3.38) can be organized into a closed algebra form. Introducing the notation ǫ, η to express the ± signs, (3.36) and (3.38) read Furthermore using the formula we can arrange the CZ ± and their mixing algebras (3.39) and (3.40) in one final compact form which we shall denote the CZ * algebra.
In closing this section we put a brief remark on the q-derivative expression. (3.31) is equivalent to another Together with (3.4), we have a q ↔ q −1 symmetric form of the standard q-derivative (with q substituted by This type of q-derivative is used in operator product expansion form of CZ + algebra [38,48].

Matrix representaions of CZ *
This section studies matrix representations of CZ * and its related algebras. We discuss the role of CZ ± commutative representations and their connection to quantum plane and * -bracket structure. We also present preliminary results about relationship of CZ * related algebras with the TBM Hamiltonian series.
To begin with, we introduce the CZ ± matrices in Section 4.1. The CZ ± operator consists of a combination of commutative and non-commutative parts, as seen in the case of MT realization. Section 4.2 considers the meaning and role of commutative representations X n and Y n , which play the same part as the MT operatorT n . The commutative representation shows its significance in connection to quantum plane picture and the * -brackets. Concrete verification is done in Section 4.4. The matrix expression for CZ * is also given in Section 4.2. Section 4.3 defines the representation sequence CZ * family, which is the sequence obtained by the replacement q → q k . We introduce some matrix representations of CZ * family associated with the TBM Hamiltonian (concrete correspondence will be verified in Section 5).
In Section 4.4, considering a special combination of DMT that provides a commutative representation of CZ ± , we derive another definition of * -product based on the quantum plane picture of TBM. The AB phase (see Section 5.1) associated with the movement of particles by DMT corresponds to the fluctuation of the quantum plane, and the phase factor generated by the successive DMT operations is expressed by a certain ordered product to reproduce the * -bracket.

cyclic representation of CZ ± algebra
In order to examine the CZ * algebra, we must first set up its generators. Matrix representations of CZ − are already known [39], but those of the other algebras are not. Thus we have to find a general expression for CZ + generators satisfying (3.36). The general expression obtained in this subsection does not always satisfy the CZ * relation (3.38). This problem is solved by using an automorphism of CZ ± in Section 4.2.
Let us follow the basic idea given in [39]. The cyclic matrix representaion of CZ generators are given if q is a root of unity The minimal set of Ln elements are as follows depending on whether N is even or odd: {L−m+1, · · · , L−1, L0, L1, · · · , Lm} for N = 2m (4.2) {L−m, · · · , L−1, L0, L1, · · · , Lm} for N = 2m + 1 . The minimal set called the "fundamental cell" [39] and the translational group GN = {G k N : n → n+kN ; k ∈ Z} on a one-dimensional lattice of period N (> 2) gives rise to an algebra automorphism of CZ algebra Ln → L n+kN , k = 0, ±1, ±2, · · · (4.4) These correspond to a magnetic unit cell and the magnetic translation group in a Bloch electron system respectively.

roles of trivial form and CZ ±
We here deal with L ± n in parallel, in order to consider the extension of CZ ± to CZ * later (see bottom of the subsection). As discussed later (Section 4.4) the phase factors of deformed commutators for L ± n correspond to translations accompanied by phases q ±1 in two orthogonal directions on a quantum plane. This is one of the reasons why we extend the CZ algebra to CZ ± in this papar.
Before discussing the CZ * extension, some remarks are in order. The first one is that the role of Q 2 as a central element [39] can be extended to Q ±2 for CZ ± algebras as follows: where these elements are related to the central elements S ± 0 (S + 0 = S0) In the form of unified expression (ǫ = ±), one may have the relations and From these, we again recognize S ± 0 ∝ Q ±2 . The second remark is about trivial (commutative) representations Two examples of gn are given in [39] gn = c n H n , or gn = q cn 2 Q 2cn H n . (for const. c) (4.14) As can be seen in (4.7), L ± n is a linear combination of trivial part H n and nontrivial part Q ±2 H n . We now recall thatL ± n is also composed of trivial (commutative)T (0) n and nontrivial (noncommutative)T (±2) n parts as previously mentioned below (3.26). In this sense the commutative representation H n (essentially X −n ) plays a key role in CZ ± algebras. It is also interesting to note that the substitution There is another trivial representation in terms of Y gn = c n Y n (4. 16) and we further notice that X n and Y n play an important role to understand a relation between quantum plane and the * -bracket (2.9) as shall be discussed in Section 4.4. There we show that the commutative representations are nontrivially realized by composite operators of DMT units in two directions on a magnetic lattice. Commuting operators again behave like noncommuting operators in the framework of quantum plane, which generates phase factors when operators are exchanged. It will turn out that the * -bracket for X n and Y n fit perfectly with this quantum plane picture in the system of TBM.
To close this subsection, we present a matrix representation of CZ * algebra based on (4.7). In order to complete the set of algebras (3.42), eq.(3.38) the remaining algebra [L + n , L − n ] * should be satisfied in addition to CZ ± algebras (3.36).
If we choose b = 0 in the matrix repreesntation (4.7) denotingL ± n , we havẽ and this does not satisfy (3.38) but a slightly different one However, applying the following transformation that keeps CZ ± unchanged we verify that (3.38) is certainly satisfied, i.e., Since CZ ± is preserved under the transfomation (4.19), L ± n also satisfy (3.36) and (3.42). Therefore the CZ * algebra is confirmed. Note that none of (4.18) and (4.20) holds for b = 0, and we only consider the b = 0 case hereafter.

Preliminary representaions to TBM
This is a preliminary section to discuss connections between CZ * algebra and TBM (tight binding model) Hamiltonain (A.11). TBM is a two-dimensional lattice model which reproduces electron's Schrödinger equation under static magnetic field in a continuum limit. Our goal (see Section 5) is to show that TBM Hamiltonians can be expressed in the CZ * generators L ± ±1 following the basic idea presented in Appendix A, however in order to get an overview at the moment we focus our attention to some representaions derived from (4.17) and (4.19) prior to detailed investigation.
These representations are relevant to three methods of finding a relationship between CZ * algebra and TBM: (i) modification of Schrödinger equations according to the structure of quantum planes, (ii) operator factorization, (iii) change of the parameter q in CZ * algebra. Modifications of the CZ * are necessary in the latter two cases.
The first representaion is given by with the choice Since TBM HamiltonianĤ is a linear combination in X ±1 and Y ±1 as seen in Appendix Â The second type of representation is for example with the choice In this case we can extract the linear TBM Hamiltonian by factoring out the operator [Z] (for details see (5.24)). In order to see the factorization, we rather employ CZ * ′ the following algebra by introducing where arbitrary representations can be applied.
The third type of candidates are not exactly the L ± n but slightly modified operatorsĽ ± n , where we consider the algebra given by Y It is obtained by the replacement q → q1 = q where the condition q N = 1 should be changed to q N 1 = 1 as well as matrix entries q in Y1 to q1. Commutation relation qXY = Y X is then changed to q1XY1 = Y1X. After all, except for the change of power in Y and Q, everything is understood as q → q1 = q 1 2 . For the sake of later conveniences, we introduce more general The modified algebra ofĽ ± n is given by the replacement q → q k in CZ * (q), that leads to a sequence of algebras CZ * (q k ), and we refer to it as CZ * family (algebras). We then have We assume Y k = Y k and q k = q k for k ≥ 2, and (4.21) multiplied by q + q −1 corresponds to the k = 2 case.
(4.29) is regarded as an exceptional case k = 1 with Y 2 1 = Y (↔ q 2 1 = q). This is the outlined strategy of how to obtain the TBM formĤ(X, Y k ; q k ) fromĽ ± n by changing the q parameter in CZ * representations. Details are explained in Section 5.2.

quantum plane and * -bracket
We discuss quantum plane picture of CZ ± algebra in the framework of discrete magnetic translationsTx andTy in TBM. Relations between the discrete magnetic translations (DMT) and the Wyle base matrices X and Y are summarized in Appendix B for the convenience.
Let us consider the matrix expressions of DMT, for example given by (B.6): As shown in Figure 1, these DMT operators describe the translations on a two-dimensional lattice (m, n) in four directions, respectively.
Consider the points A,B,C and D on the line Yj for j = n + m fixed to a constant value, and a route of successive movements by DMTs from the point A. There are two shortest ways to get to B from A, namely via Yj−1 and via Yj+1. In the case of via Yj−1, we havê and we interpret this relation that Y 2 corresponds to a movement from A to B along Yj with a phase factor q caused by the fluctuation via Yj−1. If we go from A to C via Yj−1 twice, we understand that Y 4 is the moving operator along Yj, and q 2 the fluctuation phase factor.
Similarly in the case of via Yj+1, we havê and interpret that Y 2 corresponds to a movement from A to B along Yj with a phase factor q −1 caused by the fluctuation via Yj+1. As to the movements from A to D, which is in a opposite direction to B along Yj, we understand in a parallel way that Y −2 corresponds to the movements from A to D with a phase factor q ±1 caused by the fluctuation via Yj±1.
Let us define a positive direction for Yj as the one with increasing n of the vertical axis, and denote the numbers of fluctuations via Yj+1 (resp. Yj−1) by k (resp. l) for positive direction along Yj (they are denoted by −k and −l for negative direction) when moving to an arbitrary point which is k + l points away along Yj. Then we can express the translation operator composed of k + l translations along Yj in the following way with the total phase factor q −k+l Y 2(k+l) . Then we can express (4.40) using the * -product definition as By the way, Y 2n is nothing but the trivial representation (4.13) with (4.16), and we therefore have the CZ + algebra with the definition (4.41) The commuting operator Y 2 on the line Yj acquires a nontrivial phase factor related to the * -product (4.41) as an effect of fluctuations via Yj±1. The * -product plays the function of projecting a commuting operator product into a noncommuting one. We conclude that this fact is formulated by L ′ + , which is a trivial CZ + representation. In other words, we have obtained the picture that commuting translation operators on a quantum line Yj (one-dimensional quantum plane) raise phase factors as an effect of quantum fluctuation of the quantum plane.  Figure 2: Translations on X r line (r = n − m) To complete the investigation, we have to consider another direction orthogonal to Yj. The argument is straightforward, but attention should be paid to matrix normalization in order to parallel the discussion above. We thus elaborate on the details with reference to Figure 2. Let us consider the points A,B,C and D on the line Xr for r = n − m, and two-way successive movements by DMTs from A to B via Xr±1. In the case of via Xr−1, we havê Similarly in the case of via Xr+1, we havê which means thatX −2 corresponds to a movement from A to B along Xr with a phase factor q caused by the fluctuation via Xr+1. The movements from A to D, which is in a opposite direction to B along Xr, are understood in a parallel way thatX 2 corresponds to the movements from A to D with the factor q ±1 caused by the fluctuation via Xr±1.
Let us define a positive direction for Xr as the one with increasing n on the vertical axis, and denote the numbers of fluctuations via Xr+1 (resp. Xr−1) by k (resp. l) for positive directions along Xr (they are denoted by −k and −l for negative directions) when moving to an arbitrary point which is k + l points away along Xr. Then we can express the translation operator composed of k + l translations along Xr in the following way with the total phase factor q k−lX −2(k+l) . Then we can express (4.45) using the * -product as Again,X 2n is nothing but a trivial representation (4.13) with (4.14), and we have the CZ − algebra with the definition (4.46) We therefore verify that the same picture as CZ + holds. Namely the commuting operatorX 2 on the quantum line Xr acquires a nontrivial phase factor related to the * -product (4.46) as an effect of quantum fluctuations via Xr±1. Since Xr is orthogonal to Yj, it can be said that L ′ ± n are the algebras belonging to directions orthogonal to each other.
We finally put a remark thatX −2 increases the position j by 2 along Xr, and effetive moving length of X −1 may thus amount to ∆j = 1 if one applies a dual lattice. Similarly Y 2 increases r by 2 along Yj, and thus Y −1 may effectively increase by ∆r = 1.

CZ * and TBM Hamiltonians
In Section 5, we show that the matrix representation of TBM corresponds to the Wyle representation of CZ * , which describes the Hamiltonian sequence covering various magnetic lattices. In Section 5.1, deriving the DMT algebra (exchange, fusion and circulation rules) in TBM, we comment on its relation to the quantum plane picture. In Section 5.2, we show that the TBM Hamiltonian sequence can be described using the ±1 modes of the matrix representation of the CZ * algebra family. Section 5.3 discusses extensions to general modes. The power of X corresponds to the Hamiltonian with the effective spacing of the magnetic lattice expanded from 1 to n. The power of Y corresponds to the Hamiltonian sequence that extends the quantum plane fluctuation (order of q) from 1 to k (Section 5.4). These Hamiltonians can be represented by the CZ * generators.

DMT and quantum plane inTBM
The purpose of this subsection is to verify the quantum plane picture of tight binding model (TBM) by showing that discrete magnetic translations (DMTs) satisfy the same properties as the MTA (exchange, fusion and circulation) of the continuous magnetic translations reviewed in Section 2.2. In contrast to the discussion in Section 4.4, we do not use the matrix representation of DMT. As a result of this picture, TBM can be regarded as a Hamiltonian system constructed on a quantum plane. TBM Hamiltonian is given by DMT on a two-dimensioanl lattice as follows [19,20,21,53]: the eigenvalue equation HΨ = EΨ with (5.1) is known to reduce to the following Schrödinger equation [19,21] e iθ x m−1,n ψm−1,n + e iθ y m,n−1 ψm,n−1 + e −iθ x m,n ψm+1,n + e −iθ y m,n ψm,n+1 = Eψm,n . Concerning the fusion algebra, we have to define new composite operatorTx+y satisfying the following fusion relations with phase factor ξm,n, which will be determined later TxTy |ψm,n := e iξm,nT x+y |ψm,n , (5.12) TyTx |ψm,n := e −iξm,nT x+y |ψm,n . Introducing the parameter q as we summarize the exchange, fusion and circulation as followŝ If we combine the fusion and exchange rules intô  19) is not necessarily a constant because φ, given by (5.11), depends on its site (m, n). As discussed in Section 4.4, the q can be regarded as the fluctuation of quantum plane, and it is related to the AB phases as seen in (5.11).
This interpretation suggests that the * -products are generated by the quantization of space (quantum plane) in view of discretization. It is interesting that the quantum plane picture can be understood as the underlying structure before a periodic condition is taken into account.

CZ * and TBM Hamiltonian family
Hereafter we discuss the relations between TBM HamiltonianĤ and the CZ * matrix representations L ± ±1 defined in (4.21)-(4.26). Since our three representations (4.21),(4.24) and (4.29) have different Ydependence fromĤ (see (A.11)), it is not straightforward to find their relationships. For convenience of discussion, we explicitly show the dependence on deformation parameters q and matrix sizes N in CZ * and TBM HamiltonianĤ, such as CZ * (q, N ) andĤ(q, N ′ ), where the latter TBM matrix sizes are given by N ′ = 2Q. Although the same symbols for q and N are employed in both CZ * andĤ, they are originally introduced independently, and hence generally different. Then denoting the deformation parameter of TBM by q k when both q are related, we consider the situation that the HamiltonianĤ(q k , Q) coincides withȞ k which is a linear combination of CZ * (q k , N ) generators. Keeping the relation of CZ(q k , N ) to its parent CZ(q, N ), and providing a certain relation between N and Q, we are going to determine the values of q and q k in each case. (In the case of the factorization (4.24), we do not have to consider this issue, because q k coincides with q, which is nothing but a matrix element of Y of size N = 2Q.) Let us first consider the second type representation, that is the factorized form (4.24), where the powers of Y coincide with those inĤ up to the factorization of [Z]. In this case the CZ * algebra is slightly modified to the CZ * ′ algebra, which is given by L + n and L satisfying (3.36), (4.26), (4.27). It is convenient to defineĤZ by using the CZ * ′ generators aŝ Using the relation qXY = Y X, we find that [Z] is factorized fromĤZ aŝ Next, let us consider the case (4.29), which is one of the third types and its corresponding algebra is CZ * (q1) defined in (4.32) and (4.33). DefiningȞ1 in terms of the CZ * (q1) generatorsĽ ± ±1 27) and substituting (4.29) on the r.h.s. of this, we obtain the same form as the TBM HamiltonianĤ where Y1 and q1 are substituted for Y and q in (4.23). In this representation we have the conditions X N = Y N 1 = 1 as well as Y N = 1. Both should be satisfied, and it is realized in the following way: Y1 with the relation q = q 2 1 is related to Y set in the CZ * representation (4.21) by the relation Y 2 1 = Y . Notice that it does not mean that (4.21) coincides with (4.29). If we choose N = 2Q remembering that 2Q is the matrix size ofĤ, we have as well as for Y and q where the double sign ± is introduced for a complex conjugation system. Thus we have Finally we deal with the rest of all, the first type (4.21) and the third type (4.30) in the same formalism CZ * (q k ), since (4.21) is a special case of the third type with k = 2. DefiningȞ k aš and substituting (4.30) on the r.h.s., we obtain for k = 2 where q2 = q 2 . In this representation we have the condition X N = Y N 2 = 1. Note that Y N = 1 this time. Y2 with the relation q2 = q 2 is related to Y in the CZ * representation (4.21) by the relation Y2 = Y 2 . If we as well as for Y Thus we have As to the third representation (4.30), considerig CZ * family for k ≥ 3 in the same way as above, we verify that the matrix (4.30) describes the TBM Hamiltonian familyȞ k given by (5.33) where Y k and q k are given by 40) 5.3 generalization ofĽ ± n to other modes n = ±1 As discussed in Section 4, X ±2 have the effect of increasing or decreasing j by 2 (∆j = 2) along the quantum line Xr. Reflecting this feature,Ĥ(X 2 , Y ; q) is deduced to describe the system of which effective interval ∆j is twice that ofĤ(X, Y ; q). Then denoting another TBM HamiltonianĤ(X 2 , Y ; q) byĤ2, we can derive the Schrödinger equation The new HamiltonianĤ2 possesses Uq(sl2) symmetry: However it is rather convenient to regard this symmetry as the n = ±2 parts of CZ * representation family (4.30) when considering the following Hamiltonian serieŝ which gives the Schrödinger equation with the effective interval ∆j = n = 2ν Namely, as a generalization of Section 5.2, definingȞ (n,k) in terms of the representation (4.30) of the CZ * we thus find the connection of the Hamiltonian seriesĤn to the CZ * family operatorsĽ ± n , which are extended from the n = ±1 modesĽ ± ±1 . Note that the previous casesȞ1 andȞ2 discussed in Section 5.2 belong to the n = 1 series ofȞ (n,k) (5.51)

Hamiltonian series with Y ±k family
Regarding the first type representation (4.21), it may be more convenient to consider the Hamiltonian H(X, Y 2 ; q), instead of the originalĤ(X, Y ; q). In this way, one may anticipate the avoidance of the complicated discussion in the previous subsection and a more direct correspondence between (4.21) and H(X, Y 2 ; q).
Let us consider the following Hamiltonian familŷ and first we set n = k = 2Ĥ This leads to the following Schrödinger equation 54) and it corresponds to a system whose effective interval ∆j and the size of quantum fluctuation q (see §4) are twice those of the original system (A.9), since (5.54) coincides with the equation obtained from (A.9) by the replacements ∆j = 1 → 2 and q → q 2 . It is straightforward to verify that the Hamiltonian (5.52) describes the quantum plane system with the effective interval n and the fluctuation size q k .
If we introduce the following Hn operator composed of the generators L ± n given in the CZ * representation (4.21) The HamiltonianĤ (2,2) is also given by a combination of the generators of the quantum algebra Uq(sl2) Needless to say, the CZ * representation (4.21) is again suitable to describe the relation between Hn and the Hamiltonian seriesĤ (n,2) Hn =Ĥ (n,2) =Ĥ(X n , Y2; q2) .
We also have its generalization asĤ (n,k) =Ĥ(X n , Y k ; q k ) . (5.62) As seen above,Ĥ (n,k) expresses a variety of Hamiltonians designated by combinations of effective interval of magnetic lattice ∆j and quantum plane fluctuation q k . There exists a Uq(sl2) symmetry in theĤ (n,k) system for each n and k, for example, (5.57) forĤ (2,2) , (5.43) forĤ (2,1) , and (A.15) forĤ (1,1) . On the other hand, it is possible to express the Hamiltonian series in a unified manner such as (5.50), (5.61) and (5.62), if we employ one of the closed algebra systems CZ * or CZ * family as observed in (5.49) and (5.55).

Conclusions and discussions
In this paper we have focused on the relations between CZ algebras and the quantum plane picture using algebraic properties of DMT as well as MT. The mechanism of generating phase factors is found to be compatible with the * -bracket feature of CZ algebras, and it is very convenient to investigate various properties of CZ algebras. As a result, we have clarified some properties that could not be obtained from the q-harmonic oscillators representaion (3.10). They are what mechanism determines the phase factor in (2.10), that it is related to fluctuations on the quantum plane, and that there is a certain rule in the method of constructing the CZ generators.
Commutative representation is especially important, and we need a specific pair of commuting and noncommuting operators. In the MT representation, by introducing ]the weight of MT and CZ operators, we have presented the definition of * -bracket (3.20) that can express the three types of CZ algebras, CZ ± and CZ * , in the unified form. The CZ ± operator (3.31) is a linear combination of commutativeT (0) n and noncommutativeT (±2) n operators, and the same structure is also found for the DMT matrix representation (4.7). We in fact observed in Section 4.2 that the commutative representations X n and Y n play the same role as the MT operatorT and (4.47), we recognize that the CZ ± algebras can be described by the * -ordered products (4.40) and generated by successively replacing q → q k . However, considering a single CZ * algebra with a sequence of Hamiltonians rather than the CZ * algebra family is physically easier to understand.) In this way, the CZ * algebra may be regarded as a universal algebra to describe the Hamiltonian series in accordance to various quantum plane settings of n and k.
The similarities between MT and DMT representations found in this paper may suggest a universal property common to various CZ * representations. The correspondence between the q-differential representation (3.6) and the MT representation (3.11) may reveal a physical meaning of q-differential operators in lattice systems with CZ * algebraic structure. The matrix representation of TBM Hamiltonian series by Wyle base implies the existence of quantum plane behind the physical systems. All these observations are related to the representations of CZ * algebra, and we therefore believe that significance of CZ * algebra has been increased by this paper. We will be able to clarify unsolved issues and universal properties of CZ * algebra from some properties common to multiple representations including MT and DMT representations as we have done for the question in (2.10).

A quantm group symmetry in TBM
In this appendix, we review the quantum algebra symmetry in TBM, that is, the TBM Hamiltonian is written by Uq(sl2) raising-lowering operators [19]. We also put a remark on q-inversion symmetry of the Hamiltonian. In order to see the Uq(sl2) structure, we impose a periodic condition on the Schrödinger equation (5.5), and we then transform (5.5) into a matrix form in use of the Wyle base matricis X and Y .
(A. 15) Now, let us put some remarks on q-inversion symmetry of the Hamiltonian. We use the inverted q = e iπφ till the end of this Appendix. Considering complex conjugation of (5.5), we have the Hamiltonian which looks different fromĤ. Of course,Ĥ andĤ ′ describe the same physics becauseĤ ′ is obtained by the replacement of q in (A.11) with q −1 (recall that Y (q −1 ) = Y −1 (q)). We then notice that for each expression and they satisfy the same relations as (A.14). The inverted Hamiltonian is thus given by the identical form which is written as the sum of E ′ ± the raising and lowering operators of Uq(sl2).

B matrix representations of DMT
In AppendixA, we have the matrix representationĤ (see (A.11)) of the TBM Hamiltonian H given in (5.1). This implies that there is a correspondence between the operatorsTx,Ty in (5.1) and the matrices X, Y in (A.11). In this appendix, we clarify the correspondence and then we verify the q-inversion symmetry (A.18) and the DMT algebras (5.20) in matrix representation.
In order to see the correspondence, let us consider the j-th component of matrix actions of X and Y on ψ (X ±1 ψ)j = ψj∓1 , (Y ±1 ψ)j = q ±j ψj . (B.1) These relations mean that X shifts the coordinate j by 1 and Y generates a phase factor q j .

(B.5)
Repeating the same process for the rest of DMT operators, we have the following correspondence, namely the matrix representation Also for the complex conjugate system (q = e iπφ ) with changing the mid band condition k+ ↔ −k+ and denoting the DMT operators byT ′ x ,T ′ y etc., we havê