ar X iv : m at h / 02 10 26 0 v 2 [ m at h . Q A ] 1 4 N ov 2 00 2 On representations of braid groups determined by directed graphs

ar X iv : m at h / 02 10 26 0 v 1 [ m at h . Q A ] 1 7 O ct 2 00 2 On representations of braid groups determined by directed graphs 1

We show that any Markov L-coalgebra, i.e. a k-vector space equipped with two coproducts representing a directed graph yields a solution of the Yang-Baxter equation and thus a representation of braid groups. If the directed graph is equipped with a family of probability vectors then such family yields also a representation of braid groups. We end by a generalisation of these construction to any ...

ar X iv : 0 71 1 . 23 72 v 1 [ m at h . G R ] 1 5 N ov 2 00 7 Braid groups and Artin groups

ar X iv : m at h / 06 02 50 8 v 2 [ m at h . G R ] 1 9 N ov 2 00 7 PRESENTATIONS OF FINITE SIMPLE GROUPS : A QUANTITATIVE APPROACH

There is a constant C 0 such that all nonabelian finite simple groups of rank n over Fq, with the possible exception of the Ree groups 2 G 2 (3 2e+1), have presentations with at most C 0 generators and relations and total length at most C 0 (log n + log q). As a corollary, we deduce a conjecture of Holt: there is a constant C such that dim H 2 (G, M) ≤ C dim M for every finite simple group G, e...

ar X iv : m at h / 99 11 07 4 v 1 [ m at h . Q A ] 1 1 N ov 1 99 9 CRYSTAL GRAPHS FOR BASIC REPRESENTATIONS OF THE QUANTUM AFFINE

We give a realization of crystal graphs for basic representations of the quantum affine algebra Uq(C (1) 2) in terms of new combinatorial objects called the Young walls.

ar X iv : h ep - l at / 0 01 10 26 v 2 1 3 N ov 2 00 0 1 QCD vacuum structure

Several issues related to the structure of the QCD vacuum are reviewed. We concentrate mostly on results concerning instantons and center vortices.