Groups and Lie algebras corresponding to the Yang–Baxter equations

Abstract

For a positive integer n we introduce quadratic Lie algebras tr_n, qtr_n and finitely discrete groups Tr_n, QTr_n naturally associated with the classical and quantum Yang–Baxter equation, respectively. We prove that the universal enveloping algebras of the Lie algebras tr_n, qtr_n are Koszul, and compute their Hilbert series. We also compute the cohomology rings for these Lie algebras (which by Koszulity are the quadratic duals of the enveloping algebras). Finally, we construct a basis of U(tr_n). We construct cell complexes which are classifying spaces of the groups Tr_n 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 tr_n, qtr_n map onto the associated graded algebras of the Malcev Lie algebras of the groups Tr_n, QTr_n, respectively. In the case of Tr_n, 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 Tr_n and QTr_n are not formal for n⩾4.