Quantum (non-commutative) toric geometry: Foundations

In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori classical theory replaced by quantum tori. geometry is non-commutative version theory; it generalizes non-trivially most theorems and properties geometry. By considering as (non-algebraic) stacks, show that their category equivalent to a certain fans. We develop Geometric Invariant Theory (QGIT) type construction Varieties. Unlike varieties, admit moduli define spaces, prove these spaces orbifolds and, in favorable cases, up homotopy, they complex structure.