Billiard Arrays and finite-dimensional irreducible Uq(sl2)-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 (sl 2 ) and the Lie algebra sl 2 .

Upper triangular matrices and Billiard Arrays

“…We say that flag is induced by the given basis. [15,Section 6].) Suppose that we are given two flags on V , denoted by…”

“…Let d denote the subset of R 3 consisting of the three-tuples of natural numbers whose sum is d. Thus [3]. [15,Theorem 12.7].) Suppose that we are given three totally opposite …”

“…The Billiard Array concept was introduced in [15]. This concept is closely related to the equitable presentation of U q (sl 2 ) [10,15].…”

“…The Billiard Array concept was introduced in [15]. This concept is closely related to the equitable presentation of U q (sl 2 ) [10,15]. For more information about the equitable presentation, see [1,3,[5][6][7][8][9][12][13][14]16].…”

“…(See[15, Section 6].) Suppose that we are given two flags on V , denoted by{W i } d i=0 and {W i } d i=0 .Then they are opposite if and only if W…”