The recognition theorem for graded Lie algebras in prime characteristic

Abstract

Chapter 3. The Contragredient Case 3.1. Introduction 3.2. Results on modules for three-dimensional Lie algebras 3.3. Primitive vectors in g 1 and g −1 3.4. Subalgebras with a balanced grading 3.5. Algebras with an unbalanced grading Chapter 4. The Noncontragredient Case 4.1. General assumptions and notation 4.2. Brackets of weight vectors in opposite gradation spaces 4.3. Determining g 0 and its representation on g −1 4.4. Additional assumptions 4.5. Computing weights of b − -primitive vectors in g 1 4.6. Determination of the local Lie algebra 4.7. The irreducibility of g 1 4.8. Determining the negative part when g 1 is irreducible 4.9. Determining the negative part when g 1 is reducible 4.10. The case that g 0 is abelian 4.11. Completion of the proof of the Main Theorem Bibliography

Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

“…The case p > 3 being completely investigated by Block, Wilson, Premet and Strade [42,52] (see also [1]), we double-checked the cases where p < 5. The answer of [54]∪ [50] is correct.…”

“…If n := n0 + n1 ≥ 3, then the Lie superalgebra oo (1) II (n0|n1) is simple. This Lie superalgebra has a 2|4-structure; it has no Cartan matrix.…”

“…The Lie algebras sl(2) and o(3) (1) with Cartan matrices (2) and (1), respectively, and the Lie superalgebra osp(1|2) (which is oo (2) hei(2|0) and Heisenberg superalgebra hei(0|2) sl(1|1), respectively; their (super)dimensions are 3 and 1|2, respectively.…”

“…The Lie superalgebra g(M) has a simple ideal, of sdim = 10|16 which is oo(1; 4|4) (1) /c (to be described separately below) and the quotient is isomorphic to sl(3). 2) p = 2 : The order 2 automorphisms of the orthogonal and ortho-orthogonal series give the following fixed points (recall the definition ofĝ in (6.12)): ooc(2; 2k0|2k1) for k0 + k1 odd ooc(1; 2k0|2k1) for k0 + k1 even oc(2; 2k) for k odd oc(1; 2k) for k even (15.2) 3) The following are the fixed points of order 2 automorphisms of the exceptional Lie (super)algebras for p = 3, and also, for p = 2, of the periplectic superalgebras, and of order 3 automorphisms of the orthogonal algebra and ortho-orthogonal superalgebras.…”

“…There are, however, simple Lie superalgebras with solvable even part other than oo ΠΠ (4|4), (16.11) though simple (perhaps, modulo center), have solvable even parts. One can not get a simple Lie superalgebra from oo (1) ΠΠ (2|2) passing to derived and factorizing.…”