Bialgebra actions, twists, and universal deformation formulas. (English) Zbl 0938.17015

Let \(B\) be a bialgebra and let \(F\) be an invertible element in \(F\otimes F\) satisfying \((\varepsilon\otimes\text{id})F=1=(\text{id}\otimes\varepsilon)F\) and \((\Delta\otimes\text{id})(F) (F\otimes 1)=(\otimes\Delta)(F)(1\otimes F)\). Here \(\Delta\) and \(\varepsilon\) are respectively the comultiplication and counit of \(B\). The celebrated twisting operation, which consists in replacing the original \(\Delta\) by \(\Delta_F=F\Delta F^{-1}\), had many applications to quantum groups and Hopf algebras, after its introduction in [V. G. Drinfel’d, Leningr. Math. J. 1, No. 6, 1419-1457 (1990); translation from Algebra Anal. 1, No. 6, 114-148 (1989; Zbl 0718.16033)].
In this interesting article, the authors consider the role of the twisting operation in quantization of Lie bialgebras, relating it to universal deformation formulas. As a concrete application, they establish new universal deformation associated to central extensions of Heisenberg Lie algebras, which are generalizations of the Moyal formulas.


17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)


Zbl 0718.16033
Full Text: DOI arXiv


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