Universal $q$-differential calculus and $q$-analog of homological   algebra
  We recall the definition of $q$-differential algebras and discuss some representative examples. In particular we construct the $q$-analog of the Hochschild coboundary. We then construct the universal $q$-differential envelope of a unital associative algebra and study its properties. The paper also contains general results on $d^N=0$. 
Michel Dubois-Violette and Richard Kerner