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 ${\mathfrak{sl}}_2$.

arXiv comment53 pages

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Cited by in corpus (1)

• The Lusztig automorphism of $U_q({\mathfrak{sl}}_2)$ from the equitable point of view