Deformed Commutators on Quantum Group Module-algebras

We construct quantum commutators on module-algebras of quasitriangular Hopf algebras. These are quantum-group covariant, and have generalized antisymmetry and Leibniz properties. If the Hopf algebra is triangular they additionally satisfy a generalized Jacobi identity, turning the modulealgebra into a quantum-Lie algebra. The purpose of this short communication is to present a quantum commutator structure which appears naturally on any module algebra A of a quantum group H . In section 1 we write down the main properties we require from a generalized commutator on a quantum group module-algebra, and we give its definition. In section 2 we prove a theorem collecting the main properties of this algebraic structure. Finally, in section 3 we develop an example, showing some explicit calculations for the reduced SLq(2,C) quantum plane. We refer the reader to the Appendix for notation and some basic facts on quasi-triangular Hopf-algebras. 1. The q-commutator LetH be a quasi-triangular Hopf algebra. Take A someH-module-algebra (a left one, say). As usual, we will denote the action of h ∈ H on a ∈ A by h ⊲ a, and the coproduct using the Sweedler notation ∆h = h1 ⊗ h2. Being a left-module-algebra, of course h ⊲ (ab) = (h1 ⊲ a) (h2 ⊲ b). As our main goal is to define a covariant commutator for which some generalized Leibniz rule holds on both variables, a natural way to start is proposing a deformation of the usual [a, b] = ab−ba structure valid on any associative algebra. The deformation we start with is [a , b]χ ≡ m ◦ (1− χ) (a⊗ b) (1) = ab−m(χ(a⊗ b)) Here m is the product on A and the linear map χ : A⊗A 7−→ A⊗A , which replaces the standard transposition operator τ , needs to be determined. Later on, we will sometimes use the generic decomposition (2) χ(a⊗ b) = ∑ i σ a(b)⊗ ei {ei} vector space basis of V 1991 Mathematics Subject Classification. 16W30, 16W10, 17D99 (also MSC 2000).