##
**Quasi-Hopf algebras.**
*(English.
Russian original)*
Zbl 0718.16033

Leningr. Math. J. 1, No. 6, 1419-1457 (1990); translation from Algebra Anal. 1, No. 6, 114-148 (1989).

The author defines the notion of quasi-Hopf algebra by weakening the coassociativity law for the coproduct in analogy to the weakening of the cocommutativity when passing from cocommutative to almost cocommutative Hopf algebras. A quasi-Hopf algebra is almost coassociative. The analogue of the R-matrix is an invertible element \(\Phi\) of \(A\otimes A\otimes A\), which is assumed to satisfy natural selfconsistency relations.

The paper is organized as follows: §1. Definition and elementary properties of quasi-Hopf algebras. The category mod A of left A-modules is discussed in case A denotes a quasi-bialgebra. There is an equivalence relation for quasi-bialgebras with different coproduct and different \(\Phi\), called “twisting”. §2. Quasi-Lie bialgebras. The author defines and studies the “classical analogue” or classical limit, the quasi-Lie bialgebras corresponding to quasi-Hopf quantized universal enveloping algebras. §3. Quasitriangular, triangular, and coboundary quasi-Hopf algebras. In the case of quasitriangular, triangular, and coboundary quasi-Hopf algebras the category mod \(A\) is characterized by commutativity of some diagrams. The monoidal category mod \(A\) is symmetric or a tensor category if \(A\) is triangular. Quasitriangular quasi-Hopf quantized universal enveloping algebras are discussed together with their classical limits consisting of quasi-Lie bialgebras \({\mathfrak g}\) endowed with a symmetric \({\mathfrak g}\)-invariant element \(t\) of \({\mathfrak g}\otimes {\mathfrak g}\), \(t\) vanishes for triangular quasi-Hopf algebras. The classical limit of a coboundary quasi-Hopf quantized universal enveloping algebra is proved to be a quasi-Lie bialgebra \({\mathfrak g}\) endowed with a \({\mathfrak g}\)-invariant element \(\phi\) of \(\wedge^ 3{\mathfrak g}\). \(\phi\) vanishes in the triangular case. There is stated a one-to-one correspondence up to twisting between quasi-Hopf quantized universal enveloping algebras and the quasi-Lie bialgebras (\({\mathfrak g},t)\) endowed with a symmetric \({\mathfrak g}\)-invariant tensor \(t\), i.e. it is stated, that quasi-Lie algebras of that type can be quantized (Theorem 3.15). The idea of the proof is sketched in the introduction of the paper. The paper closes with a remark on the construction of link-invariants including R-matrix invariants for classical solutions of the quantum Yang-Baxter equation.

The paper is organized as follows: §1. Definition and elementary properties of quasi-Hopf algebras. The category mod A of left A-modules is discussed in case A denotes a quasi-bialgebra. There is an equivalence relation for quasi-bialgebras with different coproduct and different \(\Phi\), called “twisting”. §2. Quasi-Lie bialgebras. The author defines and studies the “classical analogue” or classical limit, the quasi-Lie bialgebras corresponding to quasi-Hopf quantized universal enveloping algebras. §3. Quasitriangular, triangular, and coboundary quasi-Hopf algebras. In the case of quasitriangular, triangular, and coboundary quasi-Hopf algebras the category mod \(A\) is characterized by commutativity of some diagrams. The monoidal category mod \(A\) is symmetric or a tensor category if \(A\) is triangular. Quasitriangular quasi-Hopf quantized universal enveloping algebras are discussed together with their classical limits consisting of quasi-Lie bialgebras \({\mathfrak g}\) endowed with a symmetric \({\mathfrak g}\)-invariant element \(t\) of \({\mathfrak g}\otimes {\mathfrak g}\), \(t\) vanishes for triangular quasi-Hopf algebras. The classical limit of a coboundary quasi-Hopf quantized universal enveloping algebra is proved to be a quasi-Lie bialgebra \({\mathfrak g}\) endowed with a \({\mathfrak g}\)-invariant element \(\phi\) of \(\wedge^ 3{\mathfrak g}\). \(\phi\) vanishes in the triangular case. There is stated a one-to-one correspondence up to twisting between quasi-Hopf quantized universal enveloping algebras and the quasi-Lie bialgebras (\({\mathfrak g},t)\) endowed with a symmetric \({\mathfrak g}\)-invariant tensor \(t\), i.e. it is stated, that quasi-Lie algebras of that type can be quantized (Theorem 3.15). The idea of the proof is sketched in the introduction of the paper. The paper closes with a remark on the construction of link-invariants including R-matrix invariants for classical solutions of the quantum Yang-Baxter equation.

Reviewer: H.Boseck (Greifswald)

### MSC:

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

17B35 | Universal enveloping (super)algebras |