  Let G be a split, simple, simply connected, algebraic group over Q. The
degree 4, weight 2 motivic cohomology group of the classifying space BG of G is
identified with Z. We construct cocycles representing the generator of this
group, known as the second universal motivic Chern class.
  If G = SL(m), there is a canonical cocycle, defined by the first author
(1993). For any group G, we define a collection of cocycles parametrised by
cluster coordinate systems on the space of G-orbits on the cube of the
principal affine space G/U. Cocycles for different clusters are related by
explicit coboundaries, constructed using cluster transformations relating the
clusters.
  The cocycle has three components. The construction of the last one is
canonical and elementary; it does not use clusters, and provides a canonical
cocycle for the motivic generator of the degree 3 cohomology class of the
complex manifold G(C). However to lift this component to the whole cocycle we
need cluster coordinates: the construction of the first two components uses
crucially the cluster structure of the moduli spaces A(G,S) related to the
moduli space of G-local systems on S. In retrospect, it partially explains why
the cluster coordinates on the space A(G,S) should exist.
  This construction has numerous applications, including an explicit
construction of the universal extension of the group G by K_2, the line bundle
on Bun(G) generating its Picard group, Kac-Moody groups, etc. Another
application is an explicit combinatorial construction of the second motivic
Chern class of a G-bundle. It is a motivic analog of the work of
Gabrielov-Gelfand-Losik (1974), for any G. We show that the cluster
construction of the measurable group 3-cocycle for the group G(C) gives rise to
its quantum deformation.
