Lie bi-algebras on the non-commutative torus

Abstract

We use the canonical trace on the non-commutative torus to construct a Lie bi-algebra splitting of the algebra of its smooth functions. In the special case of rational parameter (the algebra is then generated by clock and shift matrices) this Lie bi-algebra is , corresponding to unitary and upper triangular matrices. The Lie bi-algebra has a remnant in the classical limit N → ∞: the elements of tend to real functions while 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.