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 $*$-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 $\mathbb{T}^2_\theta$. 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 $\underline{GL}(N)=\underline{U}(N)\oplus \underline{B}(N)$, corresponding to unitary and upper triangular matrices. The Lie bi-algebra has a remnant in the classical limit $N\to\infty$: the elements of $\underline{U}(N)$ tend to real functions while $\underline{B}(N)$ tends to a space of complex analytic functions.