Periodicity and representation type of modular Lie algebras.

1995 ◽  
Vol 1995 (464) ◽  
pp. 47-66
Keyword(s):  
Lie Algebras ◽  
Representation Type ◽  
Modular Lie Algebras

Restricted Envelopes of Lie Superalgebras

2015 ◽  
Vol 22 (02) ◽  
pp. 309-320
Author(s):  
Liping Sun ◽  
Wende Liu ◽  
Xiaocheng Gao ◽  
Boying Wu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Cartan Type ◽  
Modular Lie Algebras ◽  
Modular Lie Superalgebras

Certain important results concerning p-envelopes of modular Lie algebras are generalized to the super-case. In particular, any p-envelope of the Lie algebra of a Lie superalgebra can be naturally extended to a restricted envelope of the Lie superalgebra. As an application, a theorem on the representations of Lie superalgebras is given, which is a super-version of Iwasawa's theorem in Lie algebra case. As an example, the minimal restricted envelopes are computed for three series of modular Lie superalgebras of Cartan type.

scholarly journals On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

2010 ◽  
Vol 13 ◽  
pp. 357-369 ◽  
Author(s):  
Simon M. Goodwin ◽  
Gerhard Röhrle ◽  
Glenn Ubly
Keyword(s):  
Lie Algebras ◽  
Nilpotent Element ◽  
Coadjoint Orbit ◽  
Dimensional Representation ◽  
Simple Complex ◽  
Enveloping Algebras ◽  
Minimal Dimension ◽  
Pbw Theorem ◽  
Associated Variety ◽  
Modular Lie Algebras

AbstractWe consider the finiteW-algebraU(𝔤,e) associated to a nilpotent elemente∈𝔤 in a simple complex Lie algebra 𝔤 of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem forU(𝔤,e), we verify a conjecture of Premet, thatU(𝔤,e) always has a 1-dimensional representation when 𝔤 is of typeG2,F4,E6orE7. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal inU(𝔤) whose associated variety is the coadjoint orbit corresponding to e.

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