The action of the Cremona group on the non-commutative ring

The Cremona group acts on the field of two independent commutative variables over complex numbers. We provide a non-commutative ring that is an analog of non-commutative field of two independent variables and prove that the Cremona group embeds in the group of outer automorphisms of this ring. First proof of this result is technical, the second one is conceptual and gives a way to obtain non-commutative rings from the bounded derived categories of coherent sheaves.