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

Abstract

Abstract In this paper we prove a family of identities for Bernoulli numbers parameterized by triples of integers ( a , b , c ) ${(a,b,c)}$ with a + b + c = n - 1 ${a+b+c=n-1}$ , n ≥ 4 ${n\geq 4}$ . These identities are deduced by translating into homotopical terms the gauge action on the Maurer–Cartan set of a differential graded Lie algebra. 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.