Almost Inner Derivations of Affinizations of Minimal Q-graded Subalgebras

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.

💡 Research Summary

The paper introduces a new class of Lie algebras called minimal Q‑graded subalgebras of a semisimple Lie algebra and studies the behavior of almost‑inner derivations on them.

A Q‑graded subalgebra is defined by selecting a subset Ψ of the root system Φ such that the free abelian group Q = ℤΦ is generated by Ψ; the subalgebra then has the form L = H ⊕ ⊕_{α∈Ψ} L_α, where H is a fixed maximal torus. The subalgebra is called minimal if there is no proper Q‑graded subalgebra properly containing it. This notion differs from the usual root‑graded Lie algebras because minimal Q‑graded subalgebras are generally not perfect; they are solvable but not nilpotent.

The first main result (Theorem 2.4) shows that for any minimal Q‑graded subalgebra L, its derived algebra I =