Dual polar graphs, the quantum algebra \(U_q(\mathfrak{sl}_{2})\), and Leonard systems of dual \(q\)-Krawtchouk type

DOI10.1016/j.laa.2012.08.016zbMath1257.05184arXiv1205.2144OpenAlexW2963386927MaRDI QIDQ1932619

Chalermpong Worawannotai

Publication date: 21 January 2013

Published in: Linear Algebra and its Applications (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1205.2144

zbMATH Keywords

dual polar spacesLeonard pairs

Mathematics Subject Classification ID

Quantum groups (quantized enveloping algebras) and related deformations (17B37) Association schemes, strongly regular graphs (05E30) Graphs and abstract algebra (groups, rings, fields, etc.) (05C25)

Related Items (23)

Finite-dimensional irreducible \(U_q(\mathfrak{sl}_2)\)-modules from the equitable point of view ⋮ Compatibility and companions for Leonard pairs ⋮ The Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and \(U_q(\mathrm{sl}_2)\) ⋮ The algebra \(U_q(\mathfrak{sl}_2)\) in disguise ⋮ Some \(q\)-exponential formulas for finite-dimensional \(\square_q\)-modules ⋮ Billiard arrays and finite-dimensional irreducible \(U_q(\mathfrak{sl}_2)\)-modules ⋮ Distance-regular graphs with classical parameters that support a uniform structure: case \(q \leq 1\) ⋮ The Lusztig automorphism of Uq(𝔰𝔩2) from the equitable point of view ⋮ A \(Q\)-polynomial structure for the attenuated space poset \(\mathcal{A}_q (N, M)\) ⋮ Unnamed Item ⋮ Near-bipartite Leonard pairs ⋮ Upper triangular matrices and billiard arrays ⋮ \(Q\)-polynomial distance-regular graphs and a double affine Hecke algebra of rank one ⋮ The Terwilliger algebras of Johnson graphs ⋮ Leonard triples, the Racah algebra, and some distance-regular graphs of Racah type ⋮ Lowering-raising triples and \(U_q(\mathfrak{sl}_2)\) ⋮ The quantum adjacency algebra and subconstituent algebra of a graph ⋮ The Terwilliger algebras of Grassmann graphs ⋮ Dual polar graphs, a nil-DAHA of rank one, and non-symmetric dual \(q\)-Krawtchouk polynomials ⋮ Dual polar graphs, a nil-DAHA of rank one, and non-symmetric dual \(q\)-Krawtchouk polynomials ⋮ Uniform posets and Leonard pairs based on unitary spaces over finite fields ⋮ The attenuated space poset \(\mathcal{A}_q(N, M)\) ⋮ A note on the algebra Uq(sl2)

Cites Work

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• The universal Askey-Wilson algebra and the equitable presentation of \(U_{q}(sl_{2})\)
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• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form
• LEONARD PAIRS AND THE ASKEY–WILSON RELATIONS
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other

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