# zbMATH — the first resource for mathematics

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.

##### 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