Billiard Arrays and finite-dimensional irreducible Uq(sl2)-modules

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 Uq(sl2) and the Lie algebra sl2.