Difference operators, measuring coalgebras, and quantum group-like objects (Q1333225)

The paper studies deformed derivatives satisfying the Leibniz rule twisted by an algebra homomorphism, on a polynomial algebra and on an algebra of analytic functions of a single variable and their analogue in several dimensions. Measuring coalgebras supply the mean of generating bialgebras (Hopf algebras) from the given algebra. This method recovers the construction of the universal enveloping algebra and applied to deformed derivatives produces objects resembling quantum groups. Explicit examples for the analogue of the Lie algebra of derivations on \(\mathbb{C}^n\), the algebra of vector fields on a circle, and more general algebras of vector fields are considered.