Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lie algebras whose derivation algebras are simple

Jörg Feldvoss, Salvatore Siciliano

TL;DR

This work provides a complete description of Lie algebras whose derivation algebras are simple, unifying finite-dimensional, modular, and infinite-dimensional cases via universal central extensions. The authors show that Der$(L)$ is simple precisely when $L$ is a quotient of the universal central extension of a simple Lie algebra by a central ideal stabilized trivially by the outer derivations, linking simplicity of Der$(L)$ to covering and central-extension data. They further classify when Der$(L)$ is simple and complete, giving explicit lists in prime characteristic $p>3$ and for $\ $-graded finite-growth Lie algebras in characteristic zero, and they elucidate the structure of related covering algebras (Witt, Virasoro, Cartan types). These results extend Hochschild’s classical characterization and provide concrete structural descriptions valuable for both theory and applications in representation theory and mathematical physics.

Abstract

It is well known that a finite-dimensional Lie algebra over a field of characteristic zero is simple exactly when its derivation algebra is simple. In this paper we characterize those Lie algebras of arbitrary dimension over any field that have a simple derivation algebra. As an application we classify the Lie algebras that have a complete simple derivation algebra and are either finite-dimensional over an algebraically closed field of prime characteristic $p>3$ or $\mathbb{Z}$-graded of finite growth over an algebraically closed field of characteristic zero.

Lie algebras whose derivation algebras are simple

TL;DR

This work provides a complete description of Lie algebras whose derivation algebras are simple, unifying finite-dimensional, modular, and infinite-dimensional cases via universal central extensions. The authors show that Der is simple precisely when is a quotient of the universal central extension of a simple Lie algebra by a central ideal stabilized trivially by the outer derivations, linking simplicity of Der to covering and central-extension data. They further classify when Der is simple and complete, giving explicit lists in prime characteristic and for -graded finite-growth Lie algebras in characteristic zero, and they elucidate the structure of related covering algebras (Witt, Virasoro, Cartan types). These results extend Hochschild’s classical characterization and provide concrete structural descriptions valuable for both theory and applications in representation theory and mathematical physics.

Abstract

It is well known that a finite-dimensional Lie algebra over a field of characteristic zero is simple exactly when its derivation algebra is simple. In this paper we characterize those Lie algebras of arbitrary dimension over any field that have a simple derivation algebra. As an application we classify the Lie algebras that have a complete simple derivation algebra and are either finite-dimensional over an algebraically closed field of prime characteristic or -graded of finite growth over an algebraically closed field of characteristic zero.
Paper Structure (3 sections, 13 theorems, 18 equations)

This paper contains 3 sections, 13 theorems, 18 equations.

Table of Contents

  1. Derivations and central extensions of Cartan algebras
  2. Covering algebras
  3. Lie algebras whose derivation algebras are simple

Key Result

Proposition 1.1

Let $L$ be a Cartan algebra over a field $\mathbb{F}$ of characteristic zero. Then $L$ is complete if, and only if, $L$ is a Witt algebra $W_r$ or a contact Lie algebra $K_{2r+1}$.

Theorems & Definitions (26)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • ...and 16 more