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Quasitriangular quasi-Hopf algebras and a group closely connected to \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\). (Russian) Zbl 0718.16034
This paper is devoted to the proofs of some theorems on the structure of quasi-Hopf algebras, which were introduced before by author [Algebra Anal. 1, No. 6, 114-148 (1989; Zbl 0718.16033)]. A quasi-Hopf algebra differs from a Hopf algebra by the axiom of coassociativity which is substituted by the weaker condition: \[ (\mathrm{id}\otimes \Delta)(\Delta (a))=\Phi (\Delta \otimes \mathrm{id})(\Delta (a))\Phi^{-1},\quad a\in A, \] where \(\Phi\) is an invertible element of \(A\otimes A\otimes A\), \(A\) is a quasi-Hopf algebra, \(\Delta\) is comultiplication. The author generalizes also to quasi-Hopf algebras the notion of quasi-triangular algebra, which was introduced by him before [Proc. Int. Congr. Math., Berkeley 1986, 798–20 (1987; Zbl 0667.16003)]. So that quasi-Hopf, quasi-triangular algebra is \((A,\Delta,\varepsilon,\Phi,R)\), where \(A\) is an algebra, \(\Delta\) is comultiplication, \(\varepsilon\) is a counit, \(\Phi\in A\otimes A\otimes A\), \(R\in A\otimes A\) are invertible elements. Let \(F\in A\otimes A\) be an invertible element such that \((\mathrm{id}\otimes \varepsilon)(F)=1=(\varepsilon \otimes \mathrm{id})(F)\) holds, then one defines \({\tilde \Delta}\), \({\tilde \Phi}\), \(\tilde R\) by the following formulas: \[ \tilde \Delta(a)=F\Delta(a)F^{-1},\quad {\tilde \Phi}=F^{23}(\mathrm{id}\otimes \Delta)(F)\cdot \Phi (\Delta \otimes \mathrm{id})(F^{-1})(F^{12})^{-1},\quad \tilde R=F^{21}R\cdot F^{-1}. \] If \((A,\Delta,\varepsilon,\Phi,R)\) is a quasi-triangular quasi-Hopf algebra then it is said, that the quasi-triangular quasi-Hopf algebra \((A,{\tilde \Delta},\varepsilon,{\tilde\Phi},\tilde R)\) is got by twisting. The author considers quasi-triangular quasi-Hopf algebras, which are got by deformation on \(h\) from universal enveloping algebras. (These algebras he calls QUE-algebras.) Let \(\mathfrak g\) be a Lie algebra on \(\mathbb C[[h]]\). (\(\mathfrak g\) is a deformation of the Lie algebra \(\mathfrak g_ 0\) on \(\mathbb C\), where \(\mathfrak g_ 0=\mathfrak g/\mathfrak{hg})\). Let \(t\in\mathfrak g\otimes \mathfrak g\) be a \(\mathfrak g\)-invariant, symmetric tensor, \(A=U\mathfrak g\) is the \(h\)-adic completion of the universal enveloping algebra of \(\mathfrak g\). Let \(R=e^{ht/2}\).
Then the first main result of this paper is:
Theorem A. There exists a unique \(\Phi\) up to twisting, such that \((A,\Delta,\varepsilon,R,\Phi)\) is a quasi-triangular, quasi-Hopf algebra. A quasi-triangular, quasi-Hopf algebra, which satisfies the conditions of theorem A is called by the author a standard algebra.
Theorem A’. There exists a \(Q\)-universal formula, which expresses \(\Phi\) through \(\tau =ht\) unique up to twisting by symmetric \(Q\)-universal \(F=F(\tau)\).
Theorem B. Any quasi-triangular quasi-Hopf QUE algebra can be turned into a standard algebra by twisting.

16T05 Hopf algebras and their applications
16T20 Ring-theoretic aspects of quantum groups
17B37 Quantum groups (quantized enveloping algebras) and related deformations