The SU(2) Lie-Poisson Algebra and its Descendants

In this paper, a novel discrete algebra is presented which follows by combining the SU(2) Lie-Poisson bracket with the discrete Frenet equation. Physically, the construction describes a discrete piecewise linear string in R3. The starting point of our derivation is the discrete Frenet frame assigned at each vertix of the string. Then the link vector that connect the neighbouring vertices assigns the SU(2) Lie-Poisson bracket. Moreover, the same bracket defines the transfer matrices of the discrete Frenet equation which relates two neighbouring frames along the string. The procedure extends in a self-similar manner to an infinite hierarchy of Poisson structures. As an example, the first descendant of the SU(2) Lie-Poisson structure is presented in detail. For this, the spinor representation of the discrete Frenet equation is employed, as it converts the brackets into a computationally more manageable form. The final result is a nonlinear, nontrivial and novel Poisson structure that engages four neighbouring vertices.


I. INTRODUCTION
The Poisson structure [1] is a widely investigated concept that has both physical and mathematical relevance.The concept originates from Poisson's research on analytic mechanics, which now provides a very general and solid framework for describing Hamiltonian dynamics.Mathematically, a Poisson structure associates to every smooth function H on a smooth manifold M, a vector field X H .This vector field determines Hamilton's equation of motion, while the function H is the so-called Hamiltonian.Whenever the pertinent Poisson bracket is also a Lie bracket, it ensures the validity of Poisson's theorem that states that the Poisson bracket of two constants of motion is itself a constant of motion.
The SU(2) Lie-Poisson bracket is a classic example of a Poisson bracket structure, originally introduced by Lie [2].However, its systematic investigations came much later, and started with the seminal work by Lichnerowicz [3] who also introduced the concept of a Poisson structure.Important early contributions to the development of Poisson structures were made by Kirillov [4] and in particular by Weinstein [5] who also initiated the development of Poisson geometry (see also [6]).The concept of a Poisson structure has subsequently found numerous applications beyond the original focus that was on classical mechanics and differential geometry.Poisson structures now appear in a large variety of contexts starting from string theory, topological and conformal field theory and integrable systems [7,8]; extending to deformation quantization and non-commutative geometry; and all the way to algebraic geometry, representation theory and abstract algebra [1].
In this paper we show that a Poisson structure and in particular the SU(2) Lie-Poisson bracket can also be relevant to the development of effective theory descriptions of discrete stringlike objects.Discrete piecewise linear strings embedded in R 3 have already appeared in models of proteins, in terms of the Cα backbone [9].They have also important applications to robotics and 3D virtual reality [10].Additional applications, with more elaborate ambient manifolds, include the study of segmented string evolution in de Sitter and anti-de Sitter spaces [11]; see also [12] and [13].
The paper is arranged as follows.Initially, the descendants of the SU(2) Lie-Poisson structure that relates to the structure of a discrete piecewise linear polygonal string are considered.In addition, the model space and its reduction in the case of the standard SU(2) Lie-Poisson bracket is reviewed.Then the formalism of the discrete Frenet frames [14] and its self-similar hierarchical structure is presented.Finally, following the results of [15], the self-similar structure is converted into a spinor representation, while the Poisson brackets in terms of the SU(2) Lie-Poisson structure are introduced.That way, an infinite hierarchy of Poisson structures can be assigned to piecewise linear string as descendants of the canonical SU(2) Lie-Poisson structure.To conclude, an explicit construction of the first level descendant in this hierarchy is presented in detail.

II. THE MODEL SPACE AND THE LIE-POISSON STRUCTURE
This preparatory section summarises known results on the model space of SU(2) representations and the SU(2) Lie-Poisson structure.The starting point is a four dimensional phase space R 4 equipped with a canonical symplectic structure and Darboux coordinates (q 1 , p 1 , q 2 , p 2 ) {p α , q β } = −δ αβ , combined into two complex ones Their norm is set to be ρ, ie.
while the associated Poisson brackets have the simple form, Next define the three component unit length vector where σ a are the Pauli matrices.Then, the t a components obey the SU(2) Lie-Poisson bracket associated with the identity Therefore, ρ is a Casimir element while the phase space (1) is a model space of SU(2) representations.Note that, different values of ρ correspond to different representations.The bracket (5) determines a Poisson structure since: It is antisymmetric, ie., any two functions A and B satisfy It obeys both the Jacobi identity and the Leibnitz rule Note that the Jacobi identity coincides with the Schouten bracket of the Poisson bi-vector field from which the Leibnitz rule follows directly.
Since the rank of the antisymmetric matrix ǫ abc t c is two, the bracket in (5) does not determine a symplectic structure.However, the Poisson bracket ( 3) is symplectic with the closed and non-degenerate two-form Therefore, a Darboux coordinate representation of ( 5) can be derived by introducing the harmonic coordinates and thus, the unit length vector (4) simplifies to These coordinates foliate where (ϕ, φ, θ) are the angular coordinates and √ 2ρ the radii.That way, the symplectic two-form (11) becomes with the only non-vanishing Poisson brackets given by Finally, by setting and taking the Inönü-Wigner contraction limit (ǫ → 0) of the system (15), only the second bracket survives.The latter corresponds to the symplectic Poisson bracket on S 2 together with its closed two-form (unique up to coordinate changes), that coincides with the last term in (14).Note that the coordinate ρ appears only as a Casimir element of the Lie-Poisson bracket.Thus, for simplicity, in what follows ρ = 1.

III. DISCRETE FRENET EQUATION AND SELF-SIMILARITY
A. Vector representation of the discrete Frenet frames In this section descendants of the SU(2) Lie-Poisson bracket defined by (5), that arise in connection of open and piecewise linear polygonal strings x(s) ∈ R 3 , are constructed.To set the stage, let s be the arc length parameter with values s ∈ [0, L] while L is the length of the string.Also, V i with i = 0, ..., n are the vertices that characterise the string located at the points x(s i ) = x i .Then, neighbouring vertices are connected by the line segments and are separated by the distances The discrete Frenet frames are defined by the orthogonal triplets (t, n, b) i at the vertices V i as follows: The unit length tangent vectors t i point from V i to V i+1 the unit length binormal vectors are and the unit length normal vectors n i are computed from In addition, the transfer matrix R i+1,i maps the discrete Frenet frames between the neighbouring vertices V i and Here κ i+1 is the bond angle and τ i+1 is the torsion angle.[Note that, the transfer matrix R i+1,i ∈SO(3) engages only two of the Euler angles (κ, τ ) i since the third Euler angle becomes removed by the orthogonality of b i and t i−1 .]The torsion and bond angles (κ i , τ i ) are expressible in terms of the tangent vectors only.This observation follows directly from equation (20) since while In addition, the bond angle engages three vertices while the torsion angle engages four vertices along the string.
The aforementioned construction can be extended into an infinite hierarchy (for an infinite length string) in a self-similar manner.To do so the transfer matrix (20) is used to introduce a 2 nd level orthonormal triplet of vectors (T, N, B) i .The components of the vector T i are defined in terms of the last row of (20) while the corresponding 2 nd level binormal and normal vectors, in analogy with (18) and (19), are defined as Then the corresponding equation (20) determines the 2 nd -level transfer matrix with (K, T ) i the 2 nd -level bond and torsion angles evaluated in terms of the 2 nd -level T i in analogy to equations ( 21) and ( 22).The construction can be extended to the next level.That is, using the last row of (25) the formulation ( 23) is used to introduce the 3 rd -level tangent vectors.From these, the 3 rd level vectors (24) and transfer matrix (25) are obtained.The construction can then be continued to higher levels (in a self-similar manner) and thus, an infinite hierarchy is obtained.In particular, every vector and angle that appears in this self-similar hierarchy, can be expressed recursively in terms of the initial tangent vectors t i .

B. Spinor representation of the discrete Frenet equation
In this section the spinorial form of the discrete Frenet equation ( 20) is presented.To do so, a two component spinor is assigned to each link that connects the vertices V i and V i+1 , that is, The z i α (for α = 1, 2) are complex variables assigned to the link.Then, the unit length tangent vectors t i can be expressed in terms of the spinors from a relation akin that in (4) where σ = (σ 1 , σ 2 , σ 3 ) are the Pauli matrices, t i is the discrete tangent vector (17) and √ g i is the scale factor, The difference to equation ( 2) should be noted.From the definition (27) and using (26) one can easily derive that while in terms of the local coordinates ( 13) one obtains In analogy to (12) the value of the overall factor √ g i can be changed and let us (for simplicity) set g i = 1.
Next the conjugation operation C is introduced to create the conjugate spinor ψi , so that Together the two spinors ψ i and ψi define the 2×2 matrix where Finally, to derive the spinorial discrete Fernet equation in a matrix form, a Majorana spinor is constructed from the two spinors (26) and (31) by setting one can now introduce a spinorial transfer matrix U i+1,i that relates the Majorana spinors at the neighbouring links as Equation ( 33) is the so-called spinorial discrete Frenet equation; In analogy to (32) the matrix U i+1,i can be expressed in terms of the vertex variables Z i a (for a = 1, 2): The link (z 1 , z 2 ) i and the vertex (Z 1 , Z 2 ) i variables are connected through the discrete Frenet equation (33).In particular, and the choice In analogy with (25), one can introduce a 2 nd level spinor variables, with the ensuing 2 nd level spinorial Frenet equation.The construction can be repeated to higher levels, in a self-similar manner, to obtain an infinite hierarchy of spinorial discrete Frenet equations.Notably, all quantities that appear in this hierarchy can be written in terms of the complex variables (29), recursively.

IV. DESCENDANTS OF THE SU(2) LIE-POISSON BRACKET
In the case of the discrete Frenet frames, the entire self-similar hierarchy can be constructed recursively in terms of the initial tangent vectors (17).As a consequence, one can also introduce Poisson structures at all levels of the hierarchy; recall that the SU(2) Lie-Poisson brackets (5) imposed on the tangent vectors (17) take the simple form where ∆ i are identified as Casimir elements and for convenience the value ∆ i = 1 is chosen.Equivalently, the spinor realisation of the hierarchy can be expressed recursively in terms of the complex link variables (26).Indeed, from (36) it is straightforward to show that the link variables (29) satisfy the following algebra While it is clear that the Poisson brackets of all the quantities that appear in the self-similar hierarchy can be evaluated recursively in terms of (36) it is not obvious that the Poisson brackets of all the components of T i that appear at a given higher level of the hierarchy, form a closed algebra.If this is the case, a method is obtained to systematically generate new Poisson structures, as higher level descendants of the original SU(2) Lie-Poisson structure.In what follows, starting from the spinor representation (37) of the SU(2) Lie-Poisson bracket it is demonstrated by an explicit computation that this is the case.To do so, the Poisson brackets of the vertex variables (35) are evaluated.In particular, they are employed as coordinates to define a Poisson structure in terms of the pertinent Poisson bi-vector, that is, After some lengthy algebra it is found that the only non-vanishing brackets of the vertex variables (35) are the following where the parameter Λ i is real (ie., Λ i = Λi ) and is defined by the dual form in terms of the vertex variables either at the i th or at the i + 1 th vertex [1] .That is, Furthermore, one can check that the following identities are satisfied