The quantum algebra $U_q(s{l}_2)$ and its equitable presentation

Abstract

We show that the quantum algebra $U_q(s{l}_2)$ has a presentation with generators $x,x^{-1},y,z$ and relations $x{x}^{-1}=1$, $x^{-1}x=1$, $\frac{qxy-q^{-1}yx}{q-q^{-1}}=1$, $\frac{qyz-q^{-1}zy}{q-q^{-1}}=1$, $\frac{qzx-q^{-1}xz}{q-q^{-1}}=1$. We call this the equitable presentation. We investigate the action of $x,x^{-1},y,z$ on finite-dimensional $U_q(s{l}_2)$-modules.

Accepted for publication in J. Algebra

References in corpus (3)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• An SU(2) Analog of the Azbel--Hofstadter Hamiltonian
• An extended q-deformed su(2) algebra and the Bloch electron problem

Cited by in corpus (11)

• Poisson brackets and Poisson spectra in polynomial algebras
• Center of the universal Askey--Wilson algebra at roots of unity
• Finite-dimensional irreducible $U_q({\mathfrak{sl}}_2)$-modules from the equitable point of view
• Linear transformations that are tridiagonal with respect to the three decompositions for an LR triple
• Billiard Arrays and finite-dimensional irreducible $U_q({\mathfrak{sl}}_2)$-modules
• On the shape of a tridiagonal pair
• Evaluation modules for the $q$-tetrahedron algebra
• The algebra $U_q({\mathfrak{sl}}_2)$ in disguise
• An LR pair that can be extended to an LR triple
• The Lusztig automorphism of $U_q({\mathfrak{sl}}_2)$ from the equitable point of view
• Freidel-Maillet type presentations of $U_q(s{l}_2)$