Leonard pairs associated with the equitable generators of the quantum algebra U q ( sl 2 )

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 .

“…We now describe S 1 ∩ S 2 . By Lemma 10.7 the B-flags [1], [2] are opposite. For the moment assume r + s > N. Then S 1 ∩ S 2 = 0, so S 1 ∩ S 2 ∩ S 3 = 0 and we are done.…”

Section: More On Flagsmentioning

confidence: 94%

“…Proposition 21.14. Assume that the characteristic of F is 0 or greater than N. Then the B-flags [1], [2], [3] are, respectively,…”

Section: Examples Of Concrete Billiard Arraysmentioning

confidence: 99%

See 1 more Smart Citation

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

Terwilliger

2014

“…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].…”

Section: Introductionmentioning

confidence: 99%

Upper triangular matrices and Billiard Arrays

Yang

2016

Fix a nonnegative integer d, a field F, and a vector space V over F with dimension d + 1. Let T denote an invertible upper triangular matrix in Mat d+1 (F). Using T we construct three flags on V . We find a necessary and sufficient condition on T for these three flags to be totally opposite. In this case, we use these three totally opposite flags to construct a Billiard Array B on V . It is known that B is determined up to isomorphism by a certain triangular array of scalar parameters called the B-values. We compute these B-values in terms of the entries of T . We describe the set of isomorphism classes of Billiard Arrays in terms of upper triangular matrices.

“…(See[1, Lemma 8.3].) With reference to Definition 3.1, then there exists an irreducible U q (sl 2 )-module structure on V such that on V ,…”

mentioning

confidence: 99%

The Terwilliger algebras of Grassmann graphs

Gao

Hou

2015

Let J q (n, m) denote the Grassmann graph with vertex set X and diameter min{m, n − m}. Fix a vertex x ∈ X. Let T = T (x) denote the Terwilliger algebra of J q (n, m) corresponding to x. In this paper we study the structure of T under the assumption that m ≥ 3 and n ≥ 2m. Let U q (sl 2 ) be the quantum enveloping algebra of sl 2 and let q be the q-tetrahedron algebra. We first obtain an action of U q (sl 2 ) on the standard module of J q (n, m). Then we display a C-algebra homomorphism ϑ : U q (sl 2 ) → T and show that T is generated by the image of ϑ and some central elements of T . As an application, we also display an action of q on the standard module of J q (n, m). These results are obtained by using the theory of Leonard pairs.