Abstract
We introduce the notion of ‘bar category’ by which we mean a monoidal category equipped with additional structure formalising the notion of complex conjugation. Examples of our theory include bimodules over a *-algebra, modules over a conventional *-Hopf algebra and modules over a more general object which we call a ‘quasi-*-Hopf algebra’ and for which examples include the standard quantum groups \(u_q(\mathfrak{g})\) at q a root of unity (these are well-known not to be usual *-Hopf algebras). We also provide examples of strictly quasiassociative bar categories, including modules over ‘*-quasiHopf algebras’ and a construction based on finite subgroups H ⊂ G of a finite group. Inside a bar category one has natural notions of ‘⋆-algebra’ and ‘unitary object’ therefore extending these concepts to a variety of new situations. We study braidings and duals in bar categories and ⋆-braided groups (Hopf algebras) in braided-bar categories. Examples include the transmutation B(H) of a quasitriangular *-Hopf algebra and the quantum plane \({\mathbb C}_q^2\) at certain roots of unity q in the bar category of \(\widetilde{u_q(su_2)}\)-modules. We use our methods to provide a natural quasi-associative C *-algebra structure on the octonions \({\mathbb O}\) and on a coset example. In the Appendix we extend the Tannaka-Krein reconstruction theory to bar categories in relation to *-Hopf algebras.
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Albuquerque, H., Majid S.: Quasialgebra structure of the octonions. J. Algebra 220, 188–224 (1999)
Al-Shomrani, M.M., Beggs, E.J.: Making nontrivially associated modular categories from finite groups. Internat. J. Math. Math. Sci. 2004(42), 2231–2264 (2004)
Barr, M.: *-Autonomous Categories. Springer Lecture Notes in Mathematics 752, Berlin (1979)
Baez, J.C.: Higher-dimensional algebra. II. 2-Hilbert spaces. Adv. Math. 127(2), 125–189 (1997)
Beggs, E.J.: Making non-trivially associated tensor categories from left coset representatives. J. Pure Appl. Algebra 177, 5–41 (2003)
Bouwknegt, P., Hannabuss, K.C., Mathai, V.: Nonassociative tori and applications to T-duality. Comm. Math. Phys. 264, 41–69 (2006)
Brzezinski, T., Wisbauer, R: Corings and Comodules. LMS Lecture Notes 309, CUP (2003)
Connes, A.: Noncommutative Geometry. Academic, San Diego, CA (1994)
Drinfeld, V.G.: Quantum groups. In: Proc. ICM., AMS (1986)
Drinfeld, V.G.: Quasi Hopf algebras. Leningrad Math. J. 1, 1419–1457 (1990)
Joyal, A., Street, R.: Braided tensor categories. Adv. in Math. 102, 20–78 (1993)
Mac Lane, S.: Categories for the Working Mathematician. Springer, Berlin
Majid, S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge
Majid, S.: *-Structures on braided spaces. J. Math. Phys. 36, 4436–4449 (1995)
Majid, S.: Quasi-* structure on q-Poincaré algebras. J. Geom. Phys. 22, 14–58 (1997)
Majid, S.: Tannaka-Krein theorem for quasi-Hopf algebras and other results. Contemp. Math. 134, 219–232 (1992)
Năstăsescu, C., Raianu, Ş., Van Oystaeyen, F.: Modules graded by G-sets. Math. Z. 203, 605–627 (1990)
Năstăsescu, C., Van Oystaeyen, F., Shaoxue, L.: Graded modules over G-sets II. Math. Z. 207, 341–358 (1991)
Woronowicz, S.L.: Twisted SU(2) group. An example of a noncommutative differential calculus. Publ. Res. Inst. Math. Sci. 23(1), 117–181 (1987)
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Dedicated to Fred Van Oystaeyen, on the occasion of his sixtieth birthday.
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Beggs, E.J., Majid, S. Bar Categories and Star Operations. Algebr Represent Theor 12, 103–152 (2009). https://doi.org/10.1007/s10468-009-9141-x
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DOI: https://doi.org/10.1007/s10468-009-9141-x