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Noncommutative geometry in the sense of Connes and beyond: noncommutative algebras viewed as functions on a noncommutative space.
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$q$-Deforming Woronowicz's Leibniz Rule
At least the Drinfel'd-Jimbo quantum groups (dual to the coordinate rings) have a structure, that is usually explored by using so-called skew-derivations satisfying exactly the rule you name (see e.g. …
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Finding the Universal Ideal of a (Covariant) Differential Calculus
I don't much about differential calculi, but back in my head I also remember somewhat like this....so as a HINT (?): Could N be the kernel of something like a "quantum shuffle map" or "quantum symmetr …