Moyal Noncommutative Integrability and the Burgers - Kdv Mapping

The Moyal momentum algebra, studied in [20, 21], is once again used to discuss some important aspects of NC integrable models and 2d conformal field theories. Among the results presented, we setup algebraic structures and makes useful convention notations leading to extract non trivial properties of the Moyal momentum algebra. We study also the Lax pair building mechanism for particular examples namely, the noncommutative KdV and Burgers systems. We show in a crucial step that these two systems are mapped to each others through the following crucial mapping ∂ t 2 ֒→ ∂ t 3 ≡ ∂ t 2 ∂ x + α∂ 3 x. This makes a strong constraint on the NC Burgers system which corresponds to linearizing its associated differential equation. From the CFT's point of view, this constraint equation is nothing but the analogue of the conservation law of the conformal current. We believe that the considered mapping might help to bring new insights towards understanding the integrability of noncommutative 2d-systems.