Extended Bott-Virasoro algebra, semidirect product, *-lie algebra of diffeomorphism and noncommutative integrable systems

We study noncommutative theory of a coadjoint representation of a universal extension of Vect (S¹) ⋉ C^∞ (S¹) algebra using the action of *-deformed matrix Hill's operators Δ* on the space of *-deformed tensor densities. The centrally extended semidirect product algebra Vect(S¹) ⋉ C^∞ (S¹) is a Lie algebra of extended semidirect product of the Bott-Virasoro group Diff (S¹) ⋉ C^∞ (S¹). The study of deformed diffeomorphisms, deformed semidirect product algebra and deformed Lie derivative action of Δ* on * deformed tensor-densities on S¹ allow us to construct noncommutative two component Korteweg-de Vries (KdV) equations, in particular, we derive the noncommutative Ito equation. This leads to a geometric formulation of *-deformed quantization of the centrally extended semidirect product algebra Vect(S¹) ⋉ C^∞ (S¹) and two component noncommutative KdV equations.