The algebra $U_q({\mathfrak{sl}}_2)$ in disguise

Abstract

We discuss a connection between the algebra $U_q({\mathfrak{sl}}_2)$ and the tridiagonal pairs of $q$-Racah type. To describe the connection, let $x,y^{\pm 1},z$ denote the equitable generators for $U_q({\mathfrak{sl}}_2)$. Let $U_q^{\vee }$ denote the subalgebra of $U_q({\mathfrak{sl}}_2)$ generated by $x,y^{-1},z$. Using a tridiagonal pair of $q$-Racah type we construct two finite-dimensional $U_q^{\vee }$-modules. The constructions yield two nonstandard presentations of $U_q^{\vee }$ by generators and relations. These presentations are investigated in detail.

37 pages

References in corpus (7)

• The Universal Askey-Wilson Algebra and the Equitable Presentation of $U_q({\mathfrak{sl}}_2)$
• Some algebra related to $P$-and $Q$-polynomial association schemes
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• The equitable presentation for the quantum group $U_q(g)$ associated with a symmetrizable Kac-Moody algebra $g$
• Finite-dimensional irreducible $U_q({\mathfrak{sl}}_2)$-modules from the equitable point of view
• A classification of sharp tridiagonal pairs
• The classification of Leonard triples of QRacah type

Cited by in corpus (1)

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