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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.

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