  We introduce and begin to study Lie theoretical analogs of symplectic
reflection algebras for a finite cyclic group, which we call "cyclic double
affine Lie algebra". We focus on type A : in the finite (resp. affine, double
affine) case, we prove that these structures are finite (resp. affine,
toroidal) type Lie algebras, but the gradings differ. The case which is
essentially new involves $\mathbb{C}[u,v]$. We describe its universal central
extensions and start the study of its representation theory, in particular of
its highest weight integrable modules and Weyl modules. We also consider the
first Weyl algebra $A_1$ instead of the polynomial ring $\mathbb{C}[u,v]$, and,
more generally, a rank one rational Cherednik algebra. We study quasi-finite
highest weight representations of these Lie algebras.
