Some new simple modular Lie superalgebras

Elduque, Alberto

doi:10.2140/pjm.2007.231.337

Cited by 15 publications

(15 citation statements)

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“…Elduque [17,18,13,14] considered a particular case of the problem (9.1) and arranged the Lie (super)algebras he discovered in a Supermagic Square all its entries being of the form g(A). These Elduque and Cunha superalgebras are, indeed, exceptional ones.…”

Section: )mentioning

confidence: 99%

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Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Bouarroudj

Grozman

Leites

2009

SIGMA

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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.

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Section: )mentioning

confidence: 99%

“…Observe that although several of the exceptional examples were known for p > 2, together with one indecomposable Cartan matrix per each Lie superalgebra [17,18,13,14], the complete description of all inequivalent Cartan matrices for all the exceptional Lie superalgebras of the form g(A) and for ALL cases for p = 2 is new.…”

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confidence: 99%

Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Bouarroudj

Grozman

Leites

2009

SIGMA

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“…the first ideal Q 0 ⊗ E 1 being isomorphic to Q 0 (the simple Lie algebra of type A 1 ). As for the second ideal, the arguments in [Eld07, prove the next result:…”

Section: Gradings On Matrix Algebrasmentioning

confidence: 75%

Fine Gradings on Exceptional Simple Lie Superalgebras

2011

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“…• If > 0 and p = 1, then the analogous of Lie algebras in characteristic 0, the Brown superalgebra brj(2; 3), the Elduque superalgebra el(5; 3), the Lie superal- [29,34,40,41] for = 3, and the Brown superalgebra brj(2; 5), the Elduque superalgebra el(5; 5) [29] for = 5.…”

Section: The Weyl Groupoid Of a (Modular) Lie (Super)algebramentioning

confidence: 99%

On finite dimensional Nichols algebras of diagonal type

2017

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This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand-Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in Heckenberger (Adv Math 220:59-124, 2009) as a notable application of the notions of Weyl groupoid and generalized root system (Heckenberger in Invent Math 164: 175-188, 2006; Heckenberger and Yamane in Math Z 259:255-276, 2008). In the first part of this monograph, we give an overview of the theory of Nichols algebras of diagonal type. This includes a discussion of the notion of generalized root system and its appearance in the contexts of Nichols algebras of diagonal type and (modular) Lie superalgebras. In the second and third part, we describe for each Nichols algebra in the list of Heckenberger (2009) Communicated by Efim Zelmanov.

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Some new simple modular Lie superalgebras

Cited by 15 publications

References 15 publications

Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Fine Gradings on Exceptional Simple Lie Superalgebras

On finite dimensional Nichols algebras of diagonal type

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