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

Giaquinto, Anthony; Zhang, James J.

J. Pure Appl. Algebra 128, No. 2, 133-151 (1998).

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.

Reviewer: Nicolas Andruskiewitsch (Córdoba)

MSC:

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

Keywords:

Hopf algebras; universal deformation formulas; quantum groups

Citations:

Zbl 0718.16033

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