Lie Bi-Algebras on the Non-Commutative Torus

Infinitesimal symmetries of a classical mechanical system are usually described by a Lie algebra acting on the phase space, preserving the Poisson brackets. We propose that a quantum analogue is the action of a Lie bi-algebra on the associative $\ast$-algebra of observables. The latter can be thought of as functions on some underlying non-commutative manifold. We illustrate this for the non-commutative torus ${\mathbb{T}}_{\theta }^2$. The canonical trace defines a Manin triple from which a Lie bi-algebra can be constructed. In the special case of rational $\theta =\frac{M}{N}$ this Lie bi-algebra is $\underset{‾}{GL}(N)=\underset{‾}{U}(N)\oplus \underset{‾}{B}(N)$, corresponding to unitary and upper triangular matrices. The Lie bi-algebra has a remnant in the classical limit $N\to \mathrm{\infty }$: the elements of $\underset{‾}{U}(N)$ tend to real functions while $\underset{‾}{B}(N)$ tends to a space of complex analytic functions.