Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0406005

Braided Cyclic Cocycles and Non-Associative Geometry

S. E. Akrami·S. Majid
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 $\CC_{q}(G)$ of the standard q-deformation quantum groups. We introduce the notion of ribbon algebras in the category, which are algebras equipped with a suitable generalised automorphism $σ$, and obtain the required generalisation of cyclic cohomology. We show that this \emph{braided cyclic cocohomology} is invariant under a cochain twist. We also extend to our generalisation 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.

Explore related subjects

math.QAmath.KT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

S. E. Akrami, S. Majid. 2004-06-01. Braided Cyclic Cocycles and Non-Associative Geometry. https://doi.org/10.1063/1.1787621

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Graded Necklace Lie Bialgebras and Batalin-Vilkovisky Formalism

An involutive Lie bialgebra induces a Batalin-Vilkovisky operator on its exterior algebra. We introduce a graded generalization of the necklace Lie bialgebra, which depends on a choice of a quiver $Q$. We relate the resulting Batalin-Vilkovisky structure to the Batalin-Vilkovisky structure coming from a degree $-1$ symplectic form on a suitably defined representation variety of the quiver $Q$. The morphism intertwining these Batalin-Vilkovisky algebras will be given by a twisted trace, recovering the usual (super)trace and the odd trace.

math.QA↗
arXiv2026

Freeness and divisibility for right $H$-simple left $H$-comodule algebras over a pointed Hopf algebra $H$

Let $H$ be a pointed Hopf algebra and let $A$ be a right $H$-simple left $H$-comodule algebra. We show that every relative $(H,A)$-Hopf module is free as an $A$-module and that this freeness characterizes the class of pointed Hopf algebras. We give a criterion for the category of relative $(H,A)$-Hopf modules to be semisimple. We also show that $A$ can be embedded into a left $H$-comodule algebra of a specific form when $H$ and $A$ are $\mathbb{N}_0$-graded. As a consequence, we prove that if $H$ is finite-dimensional and $A^{\mathrm{co} H}=\Bbbk$, then $A$ is finite-dimensional and $\dim A$ divides $\dim H$.

math.QA↗
arXiv2026

$C_2$-Cofiniteness and Rationality of the Icosahedral Orbifold $V_{L_2}^{A_5}$

Let $L_2=\mathbb{Z}α$ be the rank-one root lattice with $(α,α)=2$, and let $A_5$ act on the lattice vertex operator algebra $V_{L_2}$ through an icosahedral subgroup of $\operatorname{Aut}(V_{L_2})\cong PSL_2(\mathbb{C})$. We prove that the fixed-point vertex operator algebra $V_{L_2}^{A_5}$ is strongly rational.

math.QA↗