Derivations and automorphisms of a Lie algebra of Block type

TL;DR

This paper explicitly describes the derivation algebra and automorphism group of a Lie algebra of Block type, revealing a 1-dimensional outer derivation space and a trivial inner automorphism group.

Contribution

It provides the first explicit characterization of derivations and automorphisms for the Lie algebra of Block type, including the structure of outer derivations and automorphisms.

Findings

Outer derivation space is 1-dimensional

Inner automorphism group is trivial

Explicit descriptions of derivations and automorphisms

Abstract

Let $\BB$ be the Lie algebra of Block type with basis $\{L_{\a,i}|\,\a,i\in\Z, i\geq0\}$ and relations $[L_{\a,i},L_{\b,j}]=\left((\a-1)(j+1)-(\b-1)(i+1)\right)L_{\a+\b,i+j}$. In the present paper, the derivation algebra and the automorphism group of $\BB$ are explicitly described. In particular, it is shown that the outer derivation space is 1-dimensional and the inner automorphism group of $\BB$ is trivial.