Z_n Quasialgebras

Quasialgebra structure of the octonions

We show that the octonions are a twisting of the group algebra of Z_2 x Z_2 x Z_2 in the quasitensor category of representations of a quasi-Hopf algebra associated to a group 3-cocycle. We consider general quasi-associative algebras of this type and some general constructions for them, including quasi-linear algebra and representation theory, and an automorphism quasi-Hopf algebra. Other examples include the higher 2^n-onion Cayley algebras and examples associated to Hadamard matrices.

Braided Cyclic Cocycles and Non-Associative Geometry

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 cocohomolo…

$\Z_2^n$-分次拟代数与二次型合成的 Hurwitz 问题

$\Z_2^n$-graded quasialgebras and the Hurwitz problem on compositions of quadratic forms

本文通过构造 $\Z_2^n$-分次拟代数 $\bbP_n(m)$ 推广 Clifford 代数等结构，为二次型合成的 Hurwitz 问题提供统一显式解，并得到多族新的无限解。

十六维类赛迪尼翁结合代数

Sixteen-dimensional Sedenion-like Associative Algebra

本文构造了作为$Cl_{5,0}$偶子代数的16维类赛迪尼翁结合代数，定义了其范数与对极等运算以构建Hopf代数结构，并证明了其范数保持性质及$\mathbb{Z}_2^5/2$-分级拟代数特性。

Gauge theory on nonassociative spaces

We show how to do gauge theory on the octonions and other nonassociative algebras such as `fuzzy $R^4$' models proposed in string theory. We use the theory of quasialgebras obtained by cochain twist introduced previously. The gauge theory in this case is twisting-equivalent to usual gauge theory on the underlying classical space. We give a general U(1)-Yang-Mills example for any quasi-algebra and a full description of the moduli space of flat connections in this theory for the cube $Z_2^3$ and hence for the octonions. We also obtain further results about the octonions themselves; an explicit M…