515 research outputs found
Set-theoretic solutions of the Yang-Baxter equation, Braces, and Symmetric groups
We involve simultaneously the theory of matched pairs of groups and the
theory of braces to study set-theoretic solutions of the Yang-Baxter equation
(YBE). We show the intimate relation between the notions of a symmetric group
(a braided involutive group) and a left brace, and find new results on
symmetric groups of finite multipermutation level and the corresponding braces.
We introduce a new invariant of a symmetric group , \emph{the derived
chain of ideals of} , which gives a precise information about the recursive
process of retraction of . We prove that every symmetric group of
finite multipermutation level is a solvable group of solvable length at
most . To each set-theoretic solution of YBE we associate two
invariant sequences of symmetric groups: (i) the sequence of its derived
symmetric groups; (ii) the sequence of its derived permutation groups and
explore these for explicit descriptions of the recursive process of retraction.
We find new criteria necessary and sufficient to claim that is a
multipermutation solution.Comment: 44 page
A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation
A bijective map , where
is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter
equation} (YBE) if the braid relation
holds in A non-degenerate involutive solution satisfying
, for all , is called \emph{square-free solution}. There
exist close relations between the square-free set-theoretic solutions of YBE,
the semigroups of I-type, the semigroups of skew polynomial type, and the
Bieberbach groups, as it was first shown in a joint paper with Michel Van den
Bergh.
In this paper we continue the study of square-free solutions and the
associated Yang-Baxter algebraic structures -- the semigroup , the
group and the - algebra over a field , generated by
and with quadratic defining relations naturally arising and uniquely
determined by . We study the properties of the associated Yang-Baxter
structures and prove a conjecture of the present author that the three notions:
a square-free solution of (set-theoretic) YBE, a semigroup of I type, and a
semigroup of skew-polynomial type, are equivalent. This implies that the
Yang-Baxter algebra is Poincar\'{e}-Birkhoff-Witt type algebra,
with respect to some appropriate ordering of . We conjecture that every
square-free solution of YBE is retractable, in the sense of Etingof-Schedler.Comment: 34 page
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