  For a positive integer n we introduce quadratic Lie algebras tr_n qtr_n and
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 find their Hilbert
series. We also compute the cohomology rings of these Lie algebras (which by
Koszulity are the quadratic duals of the enveloping algebras). 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. We conjecture that this map is actually an
isomorphism (this is now a theorem due to P. Lee). At the same time, we show
that the groups Tr_n and QTr_n are not formal for n>3.
