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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.

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