The PI Property in Algebras of Polynomial Type

Abstract

In this article, we study the PI property for several families of noncommutative algebras of polynomial type. Specifically, we review criteria for the PI property in double Ore extensions, two-parameter quantum Heisenberg algebras, two-parameter quantum matrix algebras, the algebra $U_q^+(B_2)$, multiparametric quantum Weyl algebras, biquadratic algebras with three generators, Noetherian Down--Up algebras, and the recently introduced algebras $B_q(f)$. In several cases, we include detailed proofs of known results, provide proofs of some identities used in the literature, and present alternative proofs of results characterizing the PI property for some of these algebras. For Noetherian Down--Up algebras, we highlight the relationship between the PI property, finiteness over the center, and the FBN property. Finally, for the algebras $B_q(f)$, we prove that they admit a PBW basis and show that the PI property can be controlled in terms of the support of the polynomial $f$.