Almost Inner Derivations of Affinizations of Minimal Q-graded Subalgebras

TL;DR

This paper studies the structure of derivations in minimal Q-graded subalgebras of semisimple Lie algebras, showing all almost inner derivations are inner and deriving formulas for loop algebra derivations.

Contribution

It introduces minimal Q-graded subalgebras, proves their derived algebras are abelian, and characterizes almost inner derivations as inner, providing a decomposition formula for derivations of affinizations.

Findings

All almost inner derivations are inner derivations.

Derived algebras of minimal Q-graded subalgebras are abelian.

Decomposition formula for derivations of loop algebras.

Abstract

Minimal Q-graded subalgebras of semisimple Lie algebras are introduced, and it is proved that their derived algebras are abelian. Almost inner derivations of minimal Q-graded subalgebras are investigated, they are all inner derivations. Based on these Lie algebras, a decomposition formula is obtained for derivations of loop algebras, and almost inner derivations of affinizations are determined.