Harmonic homogenous polynomials in 3 variables, upon
substitution of N-dimensional representations of su(2)
for the 3 commuting variables, have been used to define
a map from functions on the 2-sphere to NxN matrices,
which sends Poisson brackets into commutators
(applications ranging from M(atrix)-theory, and
integrable systems, to fluid-dynamics).
For the torus, an analogous (but very different) construction
is known. For higher-dimensional Riemann surfaces
representations of certain new, non-linear algebras, can be
found, providing a method to solve the longstanding problem
of treating different Riemann surfaces in a unified
(and concrete) way.
