Billiard Arrays and finite-dimensional irreducible $U_q({\mathfrak{sl}}_2)$-modules

Abstract

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 ${\mathfrak{sl}}_2$.

53 pages

References in corpus (4)

• The quantum algebra $U_q(s{l}_2)$ and its equitable presentation
• Dual polar graphs, the quantum algebra U_q(sl_2), and Leonard systems of dual q-Krawtchouk type
• The equitable presentation for the quantum group $U_q(g)$ associated with a symmetrizable Kac-Moody algebra $g$
• Finite-dimensional irreducible $U_q({\mathfrak{sl}}_2)$-modules from the equitable point of view