The Lie algebra of Feynman graphs gives rise to two natural representations, acting as derivations on the commutative Hopf algebra of Feynman graphs, by creating or eliminating subgraphs. Insertions and eliminations do not commute, but rather establish a larger Lie algebra of derivations which we here determine.

We say a functional equation () is stable if any function g satisfying the equation () approximatelyis near to true solution of (). Using xed point methods, we investigate approximately higherternary derivations in Banach ternary algebras via the Cauchy functional equationf(1x + 2y + 3z) = 1f(x) + 2f(y) + 3f(z) :

Let An be the free associative algebra with n generators over C, consider the Lie algebra A1 of its outer derivations (the derivations modulo the inner derivations). Let A0 be its quasi-classical limit, that it the Lie algebra of outer derivations of the free Poisson algebra with n generators over C. Boris Feigin conjectured around 1998 that the Lie algebras A0 and A1 are isomorphic. It turns o...