symmetric monoidal (∞,1)-category of spectra
The notion of a quasibialgebra generalizes that of a bialgebra Hopf algebra by introducing a nontrivial associativity coherence (Drinfeld 89) isomorphisms (representable by multiplication with an element in triple tensor product) into axioms; a quasi-Hopf algebra is a quasi-bialgebra with an antipode satisfying axioms which also involve nontrivial left and right unit coherences.
In particular, quasi-Hopf algebras may be obtained from ordinary Hopf algebras via twisting by a Drinfeld associator, i.e. a nonabelian bialgebra 3-cocycle.
Drinfel’d was motivated by study of monoidal categories in rational 2d conformal field theory (RCFT) as well as by an idea from Grothendieck‘s Esquisse namely the Grothendieck-Teichmüller tower and its modular properties. In RCFT, the monoidal categories appearing can be, by Tannaka reconstruction considered as categories of modules of Hopf algebra-like objects where the flexibility of associativity coherence in building a theory were natural thus leading to quasi-Hopf algebras.
A special case of the motivation in RCFT has a toy example of Dijkgraaf-Witten theory which can be quite geometrically explained. Namely, where the groupoid convolution algebra of the delooping groupoid of a finite group naturally has the structure of a Hopf algebra, the twisted groupoid convolution algebra of equipped with a 3-cocycle is naturally a quasi-Hopf algebra. Since such a 3-cocycle is precisely the background gauge field of the 3d TFT called Dijkgraaf-Witten theory, and hence quasi-Hopf algebras arise there (Dijkgraaf-Pasquier-Roche 91).
A quasibialgebra is a unital associative algebra with a structure of not necessarily coassociative coalgebra , with multiplicative comultiplication and counit , and an invertible element such that
(i) the coassociativity is modified by conjugation by in the sense
(ii) the following pentagon identity holds
(iii) some identities involving unit and counit hold:
It follows that .
The category of left -modules is a monoidal category, namely the coproduct is used to define the action of on the tensor product of modules , :
Using the Sweedler-like notation , formulas
define a natural transformation and the pentagon for yields the MacLane's pentagon for understood as a new associator,
For this reason, is sometimes called the associator of the quasibialgebra. While it is due to Drinfeld, another variant of it, written as a formal power series and used in knot theory is often called the Drinfeld associator (see there).
A quasi-Hopf algebra is a quasibialgebra equipped with elements and an antiautomorhphism of (a suitable kind of antipode) such that:
for with in Sweedler notation. Further we require:
The associator is a counital 3-cocycle in the sense of bialgebra cohomology theory of Majid. The 3-cocycle condition is the pentagon for . The abelian cohomology would add a coboundary of 2-cochain to get a cohomologous 3-cocycle. In nonabelian case, however, the twist by an invertible 2-cochain is done in a nonabelian way, described by Drinfeld and generalized by Majid to -cochains.
Thus, for a bialgebra , and fixed , the -th coface
for , and , . For , Majid defines
where the products are in the order of ascending . If is a cochain then its coboundary is , which is automatically an -cochain. If is an -cochain and is an -cochain then one defines a cochain twist of by by the formula
Drinfeld proved that for the following is true. Given a quasiabialgebra and a 2-cochain , the data is also a quasibialgebra. Furthermore, categories of modules and are monoidally equivalent reflecting the idea that cohomologous cocycles lead to nonessential categorical effects. If is quasitriangular quasibialgebra then we can twist the R-element to to obtain quasitriangular quasibialgebra and their braided monoidal categories of representations are braided monoidally equivalent.
Recall from Tannaka duality that given a rigid monoidal category with a fiber functor one can reconstruct a Hopf algebra via , so that in particular . While categories of the form for a quasi-Hopf algebra do not admit a fiber functor unless is furthermore a Hopf algebra, they do admit a weaker notion called a quasi-fiber functor .
Much as a fiber functor, a quasi-fiber functor is an exact, faithful functor equipped with a natural transformation
The main difference is that is not required to satisfy the associativity condition, so that does not describe a monoidal functor. The failure of to satisfy the associative condition is a direct reflection of the non-coassociativity of . See Sections 5.1, 5.11, and 5.12 in EGNO 2016 for more.
The notion was introduced in
The relation to Dijkgraaf-Witten theory appeared in
and some arguments about the general relevance of quasi-Hopf algebras is in
Recently a monograph appeared
Wikipedia article: Quasi-Hopf algebra
Other articles include
В. Г. Дринфельд, О структуре квазитреугольных квазихопфовых алгебр, Функц. анализ и его прил. 26:1 (1992), 78–80, pdf; transl. V. G. Drinfeld, Structure of quasitriangular quasi-hopf algebras, Funct. Anal. Appl., 26:1 (1992), 63–65
V. G. Drinfelʹd, О квазитреугольных квазихопфовых алгебрах и одной группе, тесно связанной с , Algebra i Analiz 2 (1990), no. 4, 149–181, pdf; translation On quasitriangular quasi-Hopf algebras and on a group that is closely connected with , Leningrad Math. J. 2 (1991), no. 4, 829–860, MR1080203
V. G. Drinfelʹd, Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, Problems of modern quantum field theory (Alushta, 1989), 1–13, Res. Rep. Phys., Springer 1989.
Shahn Majid, Quantum double for quasi-Hopf algebras, Lett. Math. Phys. 45 (1998), no. 1, 1–9, MR2000b:16077, doi, q-alg/9701002
Peter Schauenburg, Hopf modules and the double of a quasi-Hopf algebra, Trans. Amer. Math. Soc. 354 (2002), 3349–3378 pdf
M. Jimbo, H. Konno, S. Odake, J. Shiraishi, Quasi-Hopf twistors for elliptic quantum groups, Transformation Groups 4(4), 303–327 (1999) doi
Ivan Kobyzev, Ilya Shapiro, A categorical approach to cyclic cohomology of quasi-Hopf algebras and Hopf algebroids, Applied Categorical Structures, 27:1 (2019) 85–109 doi
L Frappat, D Issing, E Ragoucy, The quantum determinant of the elliptic algebra , J. Phys. A51:44, doi
See also Chapter 5, Sections 1, 11, and 12 of:
A complete classification of quasi-Hopf algebras of dimension 6:
Last revised on October 7, 2024 at 07:17:06. See the history of this page for a list of all contributions to it.