We use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a Drinfeld-type cochain twist. These are the so-called quasialgebras and include the octonions as braided-commutative but nonassociative coordinate rings, as well as quasialgebra versions of the standard -deformation quantum groups. We introduce the notion of ribbon algebras in the category, which are algebras equipped with a suitable generalized automorphism , and obtain the required generalization of cyclic cohomology. We show that this braided cyclic cohomology is invariant under a cochain twist. We also extend to our generalization the relation between cyclic cohomology and differential calculus on the ribbon quasialgebra. The paper includes differential calculus and cyclic cocycles on the octonions as a finite nonassociative geometry, as well as the algebraic noncommutative torus as an associative example.
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October 2004
Research Article|
October 01 2004
Braided cyclic cocycles and nonassociative geometry
S. E. Akrami;
S. E. Akrami
Islamic Azad University of Iran, Tehran, Iran and Department of Mathematics, University of Tehran, Tehran, Iran
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S. Majid
S. Majid
School of Mathematics, Queen Mary, University of London, London E14NS, United Kingdom
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J. Math. Phys. 45, 3883–3911 (2004)
Article history
Received:
April 28 2004
Accepted:
June 07 2004
Citation
S. E. Akrami, S. Majid; Braided cyclic cocycles and nonassociative geometry. J. Math. Phys. 1 October 2004; 45 (10): 3883–3911. https://doi.org/10.1063/1.1787621
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