[Paper Review] Classification of simple finite dimensional modular Lie superalgebras with Cartan matrix

Sofiane Bouarroudj, Pavel Grozman+1 more|arXiv (Cornell University)|Oct 26, 2007

Advanced Topics in Algebra10 citations

TL;DR

This paper classifies finite-dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices under mild technical assumptions. It identifies eleven new exceptional simple modular Lie superalgebras and reveals unique phenomena in characteristic 2, such as simple Lie superalgebras with solvable even parts and constructions from Lie algebras by oddifying Chevalley generators.

ABSTRACT

Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the corresponding Lie superalgebra is simple otherwise the quotient of the derived Lie superalgebra modulo center is simple (if its rank is greater than 1). Eleven new exceptional simple modular Lie superalgebras are discovered. Several features of classic notions, or notions themselves, are clarified or introduced, e.g., Cartan matrix, several versions of restrictedness in characteristic 2, Dynkin diagram, Chevalley generators, and even the notion of Lie superalgebra if the characteristic is equal to 2. Interesting phenomena in characteristic 2: (1) all simple Lie superalgebras with Cartan matrix are obtained from simple Lie algebras with Cartan matrix by declaring several (any) of its Chevalley generators odd; (2) there exist simple Lie superalgebras whose even parts are solvable. The Lie superalgebras of fixed points of automorphisms corresponding to the symmetries of Dynkin diagrams are also listed and their simple subquotients described.

Motivation & Objective

• To classify finite-dimensional modular Lie superalgebras with indecomposable Cartan matrices over algebraically closed fields.
• To clarify foundational notions such as Cartan matrices, restrictedness, Dynkin diagrams, and Chevalley generators in positive characteristic, especially in characteristic 2.
• To investigate the structure of Lie superalgebras when the Cartan matrix is invertible or not, particularly the simplicity of derived algebras modulo the center.
• To identify and describe Lie superalgebras arising as fixed points of diagram automorphisms, including their simple subquotients.
• To explore exceptional and non-classical behaviors in characteristic 2, such as the existence of simple Lie superalgebras with solvable even parts.

Proposed method

• Utilizes the structure theory of Lie superalgebras with Cartan matrices, focusing on indecomposable matrices and their associated root systems.
• Applies the concept of restrictedness in characteristic 2, distinguishing between multiple versions of restrictedness relevant to superalgebras.
• Employs Chevalley generators to construct Lie superalgebras from Lie algebras by assigning odd parity to selected generators.
• Analyzes the derived algebra modulo the center to determine simplicity when the Cartan matrix is non-invertible.
• Constructs fixed-point Lie superalgebras under automorphisms induced by symmetries of Dynkin diagrams and classifies their simple subquotients.
• Introduces and formalizes key structural notions—such as Dynkin diagrams and Cartan matrices—for Lie superalgebras in positive characteristic, particularly in characteristic 2.

Experimental results

Research questions

• RQ1 Which finite-dimensional modular Lie superalgebras with indecomposable Cartan matrices exist over algebraically closed fields under mild technical hypotheses?
• RQ2 How does the invertibility of the Cartan matrix affect the simplicity of the Lie superalgebra or its quotient by the center?
• RQ3 What are the structural and representation-theoretic consequences of working in characteristic 2, particularly regarding restrictedness and the behavior of Chevalley generators?
• RQ4 Can all simple Lie superalgebras with Cartan matrices in characteristic 2 be obtained by declaring some Chevalley generators of a simple Lie algebra to be odd?
• RQ5 Do there exist simple Lie superalgebras whose even parts are solvable, and if so, how are they constructed?

Key findings

• Eleven new exceptional simple modular Lie superalgebras are discovered, extending the known classification in positive characteristic.
• In characteristic 2, all simple Lie superalgebras with Cartan matrices arise from simple Lie algebras by assigning odd parity to any subset of their Chevalley generators.
• There exist simple Lie superalgebras in characteristic 2 whose even parts are solvable, a phenomenon not possible in characteristic zero.
• When the Cartan matrix is non-invertible, the quotient of the derived algebra by the center is simple, provided the rank exceeds one.
• The fixed-point Lie superalgebras under diagram automorphisms are fully listed, and their simple subquotients are explicitly described.
• Foundational notions such as restrictedness, Dynkin diagrams, and Chevalley generators are clarified and adapted for Lie superalgebras in characteristic 2.