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Bicrossproduct Hopf quasigroups

Jennifer Klim, Shahn Majid (2010)

Commentationes Mathematicae Universitatis Carolinae

We recall the notion of Hopf quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup k M k ( G ) from every group X with a finite subgroup G X and IP quasigroup transversal M X subject to certain conditions. We identify the octonions quasigroup G 𝕆 as transversal in an order 128 group X with subgroup 2 3 and hence obtain a Hopf quasigroup k G 𝕆 > k ( 2 3 ) as a particular case of our construction.

Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation

Gateva-Ivanova, Tatiana (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k {0, and z, t ∈ X is satisfied...

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