Derivations, 2-local derivations, biderivations and automorphisms of generalized Loop Heisenberg-Virasoro algebras

TL;DR

The paper studies the generalized loop Heisenberg-Virasoro algebra $\mathcal{L}(\Gamma)$, establishing a complete derivation theory, showing that all $2$-local derivations are derivations, proving that all biderivations are inner, and giving a full description of the automorphism group. It proves that $\mathrm{Der}(\mathcal{L}(\Gamma))=(\mathrm{Der}(\mathcal{L}(\Gamma)))_0+\mathrm{ad}\,\mathcal{L}(\Gamma)$ with $(\mathrm{Der}(\mathcal{L}(\Gamma)))_0$ generated by four explicit families $D_\phi$, $D_g$, $D_b$, $D^\rho$, while $\mathrm{Der}(\mathcal{L}(\Gamma))_\gamma=\mathrm{ad}$ for $\gamma\neq0$. It then shows any $2$-local derivation is a derivation and that $\mathcal{L}(\Gamma)$ is perfect, so every biderivation has the form $f(x,y)=\lambda[x,y]$, i.e., inner. The automorphism group is classified as a direct product of five types, $\mathrm{Aut}(\mathcal{L}(\Gamma))\cong A\times \mathrm{Hom}(\Gamma,\mathbb{Z})\times \chi(\Gamma)\times \mathrm{Aut}\mathbb{Z}\times \mathbb{F}^*$, with explicit automorphisms given. Overall, the work advances the structural understanding of a broad class of generalized loop Lie algebras and provides tools for symmetry analysis and representation theory in this setting.

Abstract

In this paper, the generalized Loop Heisenberg-Virasoro algebra is introduced. Firstly, we determine the derivations on the generalized Loop Heisenberg-Virasoro algebra. Then we show that all 2-local derivations are derivations. Furthermore, we determine the biderivations on the generalized Loop Heisenberg-Virasoro algebra are inner biderivations and give their applications. Finally, the automorphism groups on the generalized Loop Heisenberg-Virasoro algebra are presented.

Theorems & Definitions (41)

• Definition 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof
• Corollary 6