A New Approach to Defining Cochain Complexes for Dendriform and Pre-Lie Algebras
H. Alhussein
TL;DR
The work defines a new framework to compute cohomology of dendriform and pre-Lie algebras by embedding their cochain complexes into classical Hochschild and Lie cohomologies via tensoring with free Perm algebras. It constructs injective cochain maps that induce long exact sequences, enabling the transfer of deformation information between pre-algebraic and classical cohomologies. This reduces cohomology computations of dendriform and pre-Lie structures to well-understood Hochschild/Lie theories and clarifies structural links among Perm, pre-associative, and pre-Lie algebras. The results offer practical tools for deformation theory in these algebraic settings and highlight how free Perm algebras mediate between distinct cohomology theories.
Abstract
Our constructions provide a systematic way to study cohomology pre-algebraic structures via classical cohomology, simplifying computations and enabling the use of established techniques.