25 research outputs found
Folding of set-theoretical solutions of the Yang-Baxter equation
We establish a correspondence between the invariant subsets of a
non-degenerate symmetric set-theoretical solution of the quantum Yang-Baxter
equation and the parabolic subgroups of its structure group, equipped with its
canonical Garside structure. Moreover, we introduce the notion of a foldable
solution, which extends the one of a decomposable solution
A note on Garside monoids and Braces
A left brace is a triple , where is
an abelian group, is a group, and there is a
left-distributivity-like axiom that relates between the two operations in
. In analogy with a left brace, we define a left
-brace to be a triple , where
is a commutative monoid, is a monoid,
and the axiom of left distributivity holds. A lcm-monoid is a
left-cancellative monoid such that is the unique invertible element in ,
and every pair of elements in admit a lcm with respect to
left-divisibility. The class of lcm-monoids contains the Gaussian,
quasi-Garside and Garside monoids. We show that every lcm-monoid induces a left
-brace. Furthermore, we show that every Gaussian group induces a
partial left brace.Comment: 12 pages, 5 figure