  We introduce the notion of a Billiard Array. This is an equilateral
triangular array of one-dimensional subspaces of a vector space $V$, subject to
several conditions that specify which sums are direct. We show that the
Billiard Arrays on $V$ are in bijection with the 3-tuples of totally opposite
flags on $V$. We classify the Billiard Arrays up to isomorphism. We use
Billiard Arrays to describe the finite-dimensional irreducible modules for the
quantum algebra $U_q(\mathfrak{sl}_2)$ and the Lie algebra $\mathfrak{sl}_2$.

Quantum Algebra;Representation Theory