Lie bi-algebras on the non-commutative torus

We use the canonical trace on the non-commutative torus T-theta(2) to construct a Lie bi-algebra splitting of the algebra of its smooth functions. In the special case of rational parameter theta = M/N (the algebra is then generated by clock and shift matrices) this Lie bi-algebra is GL(N) = U(N) circle plus B(N), corresponding to unitary and upper triangular matrices. The Lie bi-algebra has a remnant in the classical limit N -> infinity: the elements of U(N) tend to real functions while B(N) tends to a space of complex analytic functions. The limit results into an infinite dimensional Lie bi-algebra for the smooth functions on the commutative torus.