Twisted Traces and Positive Forms on Generalized 𝑞-Weyl Algebras

Abstract

Let 𝒜 be a generalized 𝑞-Weyl algebra, it is generated by 𝑢, 𝑣, 𝛧, 𝛧⁻¹ with relations 𝛧𝑢𝛧⁻¹ = 𝑞²𝑢, 𝛧𝑣𝛧⁻¹ = 𝑞⁻²𝑣, 𝑢𝑣 = 𝑃(𝑞⁻¹𝛧), 𝑣𝑢 = 𝑃(𝑞𝛧), where 𝑃 is a Laurent polynomial. A Hermitian form (⋅,⋅) on 𝒜 is called invariant if (𝛧𝑎, 𝑏) = (𝑎, 𝑏𝛧⁻¹), (𝑢𝑎, 𝑏) = (𝑎, 𝑠𝑏𝑣), (𝑣𝑎, 𝑏) = (𝑎, 𝑠⁻¹𝑏𝑢) for some s ∈ C with |𝑠| = 1 and all 𝑎, 𝑏 ∈ 𝒜. In this paper, we classify positive definite invariant Hermitian forms on generalized 𝑞-Weyl algebras.