Billiard arrays and finite-dimensional irreducible \(U_q(\mathfrak{sl}_2)\)-modules. (English) Zbl 1352.17021
Summary: 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 \(\mathfrak{sl}_2\).
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 15A21 | Canonical forms, reductions, classification |
References:
| [1] | Alnajjar, H., Leonard pairs associated with the equitable generators of the quantum algebra \(U_q(sl_2)\), Linear Multilinear Algebra, 59, 1127-1142 (2011) · Zbl 1260.17011 |
| [2] | Benkart, G.; Terwilliger, P., The equitable basis for \(sl_2\), Math. Z., 268, 535-557 (2011) · Zbl 1277.17013 |
| [3] | Funk-Neubauer, D., Bidiagonal pairs, the Lie algebra \(sl_2\), and the quantum group \(U_q(sl_2)\), J. Algebra Appl., 12, 1250207 (2013), 46 pp. · Zbl 1280.17016 |
| [4] | Hartwig, B.; Terwilliger, P., The tetrahedron algebra, the Onsager algebra, and the \(sl_2\) loop algebra, J. Algebra, 308, 840-863 (2007) · Zbl 1163.17026 |
| [5] | Huang, H., The classification of Leonard triples of QRacah type, Linear Algebra Appl., 436, 1442-1472 (2012) · Zbl 1238.15002 |
| [6] | Humphreys, J. E., Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math., vol. 9 (1972), Springer: Springer New York · Zbl 0254.17004 |
| [7] | Ito, T.; Rosengren, H.; Terwilliger, P., Evaluation modules for the \(q\)-tetrahedron algebra, Linear Algebra Appl., 451, 107-168 (2014) · Zbl 1294.17014 |
| [8] | Ito, T.; Terwilliger, P., Tridiagonal pairs and the quantum affine algebra \(U_q(\hat{s l}_2)\), Ramanujan J., 13, 39-62 (2007) · Zbl 1128.16028 |
| [9] | Ito, T.; Terwilliger, P., Two non-nilpotent linear transformations that satisfy the cubic \(q\)-Serre relations, J. Algebra Appl., 6, 477-503 (2007) · Zbl 1179.17017 |
| [10] | Ito, T.; Terwilliger, P., The \(q\)-tetrahedron algebra and its finite-dimensional irreducible modules, Comm. Algebra, 35, 3415-3439 (2007) · Zbl 1133.17011 |
| [11] | Ito, T.; Terwilliger, P.; Weng, C., The quantum algebra \(U_q(sl_2)\) and its equitable presentation, J. Algebra, 298, 284-301 (2006) · Zbl 1090.17004 |
| [12] | Jantzen, J., Lectures on Quantum Groups, Grad. Stud. Math., vol. 6 (1996), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0842.17012 |
| [13] | Roman, S., Lattices and Ordered Sets (2008), Springer: Springer New York · Zbl 1154.06001 |
| [14] | Terwilliger, P., The equitable presentation for the quantum group \(U_q(g)\) associated with a symmetrizable Kac-Moody algebra \(g\), J. Algebra, 298, 302-319 (2006) · Zbl 1106.17021 |
| [15] | Terwilliger, P., The universal Askey-Wilson algebra and the equitable presentation of \(U_q(sl_2)\), SIGMA, 7, 099 (2011), 26 pp. · Zbl 1244.33016 |
| [16] | Terwilliger, P., Finite-dimensional irreducible \(U_q(sl_2)\)-modules from the equitable point of view, Linear Algebra Appl., 439, 358-400 (2013) · Zbl 1354.17012 |
| [17] | Worawannotai, C., Dual polar graphs, the quantum algebra \(U_q(sl_2)\), and Leonard systems of dual \(q\)-Krawtchouk type, Linear Algebra Appl., 438, 443-497 (2013) · Zbl 1257.05184 |
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