Arithmetic hom-Lie algebras, twisted derivations and non-commutative arithmetic schemes

TL;DR

This paper explores hom-Lie algebras, especially twisted derivations, and their potential applications in number theory and arithmetic geometry, highlighting their natural occurrence in various mathematical contexts.

Contribution

It introduces the study of hom-Lie algebras and twisted derivations within arithmetic and geometric frameworks, proposing their relevance in number theory.

Findings

Hom-Lie algebras generalize Lie algebras via twisted Jacobi identity.

Twisted derivations appear naturally in number theory and arithmetic geometry.

Potential applications of these structures in p-adic Hodge theory and Iwasawa theory.

Abstract

Hom-Lie algebras are non-associative algebras generalizing Lie algebras by twisting the Jacobi identity by an endomorphism. The main examples are algebras of twisted derivations (i.e., linear maps with a generalized Leibniz rule). Such generalized derivations seem to pop up in different guises in many parts of number theory and arithmetic geometry. In fact, any place something like $\mathrm{id}-\phi$, where $\phi$ is (possibly extended to) a ring morphism, appears, such as in $p$-adic Hodge theory, Iwasawa theory, e.t.c., there is a twisted derivation hiding. Therefore, hom-Lie algebras appear to have a natural r\^ole to play in many number-theoretical disciplines. This paper is a first step in a study of these operators and associated algebras in an arithmetic-/geometric context.