Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 060, 63 pages      arXiv:0710.5149
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Sofiane Bouarroudj a, Pavel Grozman b and Dimitry Leites c
a) Department of Mathematics, United Arab Emirates University, Al Ain, PO. Box: 17551, United Arab Emirates
b) Equa Simulation AB, Stockholm, Sweden
c) Department of Mathematics, University of Stockholm, Roslagsv. 101, Kräftriket hus 6, SE-106 91 Stockholm, Sweden

Received September 17, 2008, in final form May 25, 2009; Published online June 11, 2009

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.

Key words: modular Lie superalgebra, restricted Lie superalgebra; Lie superalgebra with Cartan matrix; simple Lie superalgebra.

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