Properties of a 29-dimensional simple Lie algebra of characteristic three

Brown, Gordon C.

doi:10.1007/bf01457452

Cited by 21 publications

(19 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For p > 0, Weisfeiler and Kac [54] gave a classification, but although the idea of their proof is OK, the paper has several gaps and vague notions (the Brown algebra br(3) was missed, whereas Brown [9] who discovered it did not write that it possesses Cartan matrix, actually two inequivalent matrices first observed by Skryabin [50,30]; the notion of the Lie algebra with Cartan matrix nicely formulated in [28] was not properly developed at the time [54] was written; the Dynkin diagrams mentioned there were not defined at all in the modular case; the algebras g(A) and g(A) (i) /c were sometimes identified). 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.…”

Section: Step 1: An Overview Of Known Resultsmentioning

confidence: 99%

Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Bouarroudj

Grozman

Leites

2009

SIGMA

View full text Add to dashboard Cite

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.

show abstract

Section: Step 1: An Overview Of Known Resultsmentioning

confidence: 99%

Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Bouarroudj

Grozman

Leites

2009

SIGMA

View full text Add to dashboard Cite

show abstract

“…• If > 0 and p = 1, then the analogous of Lie algebras in characteristic 0, the Brown algebras br(2; a), br(2), br(3) [29,31,81] for = 3, and the Kac-Weisfeiler algebras wk(3; a), wk(4; a) [29,59] for = 2.…”

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

confidence: 99%

On finite dimensional Nichols algebras of diagonal type

2017

View full text Add to dashboard Cite

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.

show abstract

“…• l = 3 and the characteristic of k is 3, and then g is isomorphic to the 29-dimensional simple Lie algebra discovered by Brown [1982], or…”

Section: Type Bmentioning

confidence: 99%