Biderivations, local and 2-local derivation and automorphism of simple $ω$-Lie algebras

TL;DR

This work extends classical results on local and 2-local derivations and automorphisms from Lie algebras to simple $ω$-Lie algebras, and provides a detailed treatment of biderivations and $\frac{1}{2}$-derivations in low-dimensional cases. Using foundational results and case-by-case analysis, it proves that local and 2-local derivations of finite-dimensional semisimple $ω$-Lie algebras are derivations, and that local automorphisms are either automorphisms or anti-automorphisms with 2-local automorphisms being automorphisms. It also supplies complete characterizations of biderivations for 4-dimensional $ω$-Lie algebras (with two exceptions) and shows that $\frac{1}{2}$-derivations in simple $ω$-Lie algebras are necessarily scalar multiples of the identity, including their 2-local variants. These results generalize known Lie‑theoretic phenomena to the broader $ω$‑Lie setting and pave the way for systematic analysis in higher dimensions and related algebraic structures.

Abstract

Given a finite-dimensional complex simple $ω$-Lie algebras $\mathfrak{}$ over $\mathbb{C}$. We prove that every local ,$2-$local derivation is a derivation and every local (resp. 2-local) automorphisms are automorphisms or an anti-automorphis (resp. automorphism). We characterize also biderivation, $\frac{1}{2}$-derivation and local (2-local) $\frac{1}{2}$-derivation of $\mathfrak{g}$.

Theorems & Definitions (56)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.1
• Proposition 2.2
• proof
• Theorem 2.1
• proof
• Lemma 2.1
• proof