The equitable q-deformation of 𝔰𝔩2

In this paper, we investigate the algebra [Formula: see text], which is known as the equitable [Formula: see text]-deformation of [Formula: see text]. This algebra was introduced in 2005 by Ito et al. Assume that [Formula: see text] is a primitive [Formula: see text]th root of unity with [Formula: see text]. We prove that [Formula: see text] becomes a Polynomial Identity (PI) algebra. It was previously known for such algebras, the simple modules are finite-dimensional with dimension at most the PI degree. We determine the PI degree of [Formula: see text], and we classify up to isomorphism the simple [Formula: see text]-modules. We also find the center of [Formula: see text].