Non-commutative Poisson Algebra Structures On Extended Affine Lie Algebras (?) Over The Quantum Torus

Non-commutative Poisson algebras are the algebras having both an associative algebrastructure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on the extended affine Lie algebras (?) of typeA, which are coordinated by the quantum torus with nullity ν, are determined. It is shownthat the associative products on the derived Lie subalgebras are trivial. The associativeproducts of the scaling elements are described as well.