The gauge action, DG Lie algebra and identities for Bernoulli numbers

In this paper we prove a family of identities for Bernoulli numbers parameterized by triples of integers $(a,b,c)$ with $a+b+c=n-1$, $n\ge 4$. These identities are deduced while translating into homotopical terms the gauge action on the Maurer Cartan Set which can be seen an abstraction of the behaviour of gauge infinitesimal transformations in classical gauge theory. We show that Euler and Miki's identities, well known and apparently non related formulas, are linear combinations of our family and they satisfy a particular symmetry relation.