A family of algebraic operations extending the Turaev cobracket

We introduce an algebraic extension of the Turaev cobracket as a family of maps parametrised by a positive integer. It is defined by constructing the generalised version of the divergence maps for an arbitrary associative algebra with the help of a connection in non-commutative geometry, and we recover the above extension by applying the construction to the case of the group algebra of the fundamental group of a compact connected oriented surface with boundary. If the connection is flat, the generalised divergence maps define classes in the Lie algebra cohomology of the space of derivations, and in the case of the free associative algebra, we show that they are canonically identified with the standard generators of the cohomology ring of the matrix Lie algebra \mathfrak{gl}_{n} .