Mathematical billiards is much like the real game: a point mass, representing the ball, rolls in a straight line on a (perfectly friction-less) table, striking the sides according to the law of reflection. A billiard trajectory is then completely characterized by the number of elastic collisions. The rules of mathematical billiards may be simple, but the possible behaviours of billiard trajectories are endless. In fact, several fundamental theory questions in mathematics can be recast as billiards problems. A billiard trajectory is called a periodic orbit if the number of distinct collisions in the trajectory is finite. We show that periodic orbits on such billiard tables cannot have an odd number of distinct collisions. We classify all possible equivalence classes of periodic orbits on square and rectangular tables. We also present a connection between the number of different equivalence classes and Euler’s totient function, which for any positive integer N, counts how many positive integers smaller than N share no common divisor with N other than $1$. We explore how to construct periodic orbits with a prescribed (even) number of distinct collisions and investigate properties of inadmissible (singular) trajectories, which are trajectories that eventually terminate at a vertex (a table corner).

This article focuses on the representation theory of algebras associated with $\mathfrak {sl}_2$, including the affine Lie algebra $\widehat {\mathfrak {sl}_2}$, the affine Kac–Moody algebra $\widetilde {\mathfrak {sl}_2}$, and the affine-Virasoro algebra $\mathfrak {Vir}\ltimes \widehat {\mathfrak {sl}_2}$. First, we classify certain modules over these algebras, which are free of rank one when restricted to some specific subalgebras. We demonstrate a connection between these modules and modules over the Weyl algebras, which allows us to construct large families of modules that are free of arbitrary finite rank when restricted to the Cartan subalgebra. We then investigate the simplicity of these modules. For reducible modules, we fully characterize their composition factors. Through a comparison with existing simple modules in the literature, we have identified a novel family of simple modules over the affine Kac–Moody algebra $\widetilde {\mathfrak {sl}_2}$. Finally, we turn our attention to a class of tensor product modules over the affine-Virasoro algebra $\mathfrak {Vir}\ltimes \widehat {\mathfrak {sl}_2}$. We derive a necessary and sufficient condition for the simplicity of these modules and determine their isomorphism classes.