414 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
Noncommutative Multi-Instantons on R^{2n} x S^2
Generalizing self-duality on R^2 x S^2 to higher dimensions, we consider the
Donaldson-Uhlenbeck-Yau equations on R^{2n} x S^2 and their noncommutative
deformation for the gauge group U(2). Imposing SO(3) invariance (up to gauge
transformations) reduces these equations to vortex-type equations for an
abelian gauge field and a complex scalar on R^{2n}_\theta. For a special
S^2-radius R depending on the noncommutativity \theta we find explicit
solutions in terms of shift operators. These vortex-like configurations on
R^{2n}_\theta determine SO(3)-invariant multi-instantons on R^{2n}_\theta x
S^2_R for R=R(\theta). The latter may be interpreted as sub-branes of
codimension 2n inside a coincident pair of noncommutative Dp-branes with an S^2
factor of suitable size.Comment: 1+8 pages, v2: reference added, version published in PL
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