1304.4097 (Ruggero Bandiera)

Non-abelian higher derived brackets    [PDF]

Ruggero Bandiera

Let $M$ be a graded Lie algebra that splits, as a graded space, in the direct sum of graded Lie subalgebras $L$ and $A$, with $A$ abelian. Let $D$ be a degree one derivation of $M$ such that $D^2=0$ and $D(L)\subset L$, then Voronov's construction of higher derived brackets associates to $D$ a $L_\infty$-structure on $A[-1]$. It has already been observed, and it follows from the results of this paper, that the resulting $L_\infty$-algebra is a weak model for the homotopy fiber of the inclusion of differential graded Lie algebras $i:(L,D,[\cdot,\cdot])\rh (M,D,[\cdot,\cdot])$. We prove this result using standard homology perturbation theory, in this way we also extend the construction when the assumption $A$ abelian is dropped: the resulting formulas involve Bernoulli numbers. Finally we apply the developed theory to recover a number of results scattered in the literature by Bering, Cattaneo and Sch\"{a}tz, Chuang and Lazarev, Getzler.

View original: http://arxiv.org/abs/1304.4097