  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.
