Cohomology and automorphisms of Com-PreLie algebras
Tao Zhang, Ying-Hua Lu
TL;DR
This work develops representations and a cohomology theory for Com-PreLie algebras $A$ and leverages them to study abelian extensions and the inducibility of automorphisms. It introduces a mixed Harrison–Dzhumadil'daev cohomology $H^\bullet(A,V)$ for a representation $V$, and shows that abelian extensions are classified by $H^2(A,V)$ with explicit cocycles $(\phi,\psi)$; it then formulates Wells maps to characterize inducibility, proving that inducibility is equivalent to the vanishing of a Wells cohomology class. A Wells-type exact sequence is established, linking $\mathcal{Z}^1(A,V)$, Aut_V($\widehat{A}$), a compatibility group $\mathcal{C}$, and $H^2(A,V)$, thereby providing a robust obstruction-theoretic framework for extensions and automorphism inducibility in the Com-PreLie setting. The results yield a coherent cohomological perspective on the structure of Com-PreLie algebras with potential applications to rooted-tree combinatorics and control-theoretic operators.
Abstract
This paper introduces the concept of representations for Com-PreLie algebras and develops corresponding cohomology theories, examining how cohomology groups can be applied in the context of Com-PreLie algebras. Initially, we utilize the cohomology theory to investigate abelian extensions of Com-PreLie algebras. Next, given an abelian extension of Com-PreLie algebras and its representation, we explore the inducibility of Com-PreLie automorphisms, deriving both necessary and sufficient conditions for the inducibility problem. Lastly, we delve deeper into the inducibility of Com-PreLie automorphisms using the Wells exact sequences, offering a clear framework for studying the inducibility of Com-PreLie automorphisms.