The Yang - Baxter Equations Laurent Bartholdi

For a positive integer n we introduce quadratic Lie algebras trn qtr n and discrete groups Trn, QTr n naturally associated with the classical and quantum Yang-Baxter equation, respectively. We prove that the universal enveloping algebras of the Lie algebras trn, qtr n are Koszul, and compute their Hilbert series. We also compute the cohomology rings of these Lie algebras (which by Koszulity are the quadratic duals of the enveloping algebras). Finally, we construct a basis of U (trn). We construct cell complexes which are classifying spaces of the groups Trn and QTr n , and show that the boundary maps in them are zero, which allows us to compute the integral cohomology of these groups. We show that the Lie algebras trn, qtr n map onto the associated graded algebras of the Malcev Lie algebras of the groups Trn, QTr n , respectively. In the case of Trn, we use quantization theory of Lie bialgebras to show that this map is actually an isomorphism. At the same time, we show that the groups Trn and QTr n are not formal for n ≥ 4.