scholarly journals On the Number of Simple Modules of a Supersolvable Restricted Lie Algebra

2000 ◽  
Vol 230 (2) ◽  
pp. 319-333
Author(s):  
Jörg Feldvoss
Keyword(s):  
Lie Algebra ◽  
Simple Modules ◽  
Restricted Lie Algebra

scholarly journals Vector bundles associated to Lie algebras

2016 ◽  
Vol 2016 (716) ◽  
Author(s):  
Jon F. Carlson ◽  
Eric M. Friedlander ◽  
Julia Pevtsova
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Bundles ◽  
Coherent Sheaves ◽  
Restricted Lie Algebra ◽  
Finite Dimensional

AbstractWe introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra

scholarly journals Complexity for modules over the classical Lie superalgebra

2012 ◽  
Vol 148 (5) ◽  
pp. 1561-1592 ◽  
Author(s):  
Brian D. Boe ◽  
Jonathan R. Kujawa ◽  
Daniel K. Nakano
Keyword(s):  
Lie Algebra ◽  
Lie Superalgebra ◽  
Projective Resolution ◽  
Lie Superalgebras ◽  
Simple Modules ◽  
Minimal Projective Resolution ◽  
Finite Dimensional ◽  
Reductive Lie Algebra ◽  
Completely Reducible ◽  
Kac Modules

AbstractLet ${\Xmathfrak g}={\Xmathfrak g}_{\zerox }\oplus {\Xmathfrak g}_{\onex }$ be a classical Lie superalgebra and let ℱ be the category of finite-dimensional ${\Xmathfrak g}$-supermodules which are completely reducible over the reductive Lie algebra ${\Xmathfrak g}_{\zerox }$. In [B. D. Boe, J. R. Kujawa and D. K. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN (2011), 696–724], we demonstrated that for any module M in ℱ the rate of growth of the minimal projective resolution (i.e. the complexity of M) is bounded by the dimension of ${\Xmathfrak g}_{\onex }$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\Xmathfrak {gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.

scholarly journals A Note on the Rank of a Restricted Lie Algebra

2018 ◽  
Vol 25 (04) ◽  
pp. 653-660
Author(s):  
Hao Chang
Keyword(s):  
Lie Algebra ◽  
Short Note ◽  
Restricted Lie Algebra ◽  
Irreducible Modules ◽  
Contact Algebra

In this short note, we study the rank of a restricted Lie algebra (𝔤, [p]), and give some applications which concern the dimensions of non-trivial irreducible modules. We also compute the rank of the restricted contact algebra.

scholarly journals Graded modules over simple Lie algebras

2019 ◽  
Vol 53 (supl) ◽  
pp. 45-86
Author(s):  
Yuri Bahturin ◽  
Mikhail Kochetov ◽  
Abdallah Shihadeh
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Arbitrary Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Simple Lie Algebras ◽  
Simple Modules ◽  
Graded Modules ◽  
Finite Dimensional ◽  
Necessary And Sufficient

The paper is devoted to the study of graded-simple modules and gradings on simple modules over finite-dimensional simple Lie algebras. In general, a connection between these two objects is given by the so-called loop construction. We review the main features of this construction as well as necessary and sufficient conditions under which finite-dimensional simple modules can be graded. Over the Lie algebra sl2(C), we consider specific gradings on simple modules of arbitrary dimension.

scholarly journals IRREDUCIBLE REPRESENTATIONS OF THE HAMILTONIAN ALGEBRAH(2r;n)

2011 ◽  
Vol 90 (3) ◽  
pp. 403-430 ◽  
Author(s):  
YU-FENG YAO ◽  
BIN SHU
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Irreducible Representations ◽  
Full Description ◽  
Closed Field ◽  
Maximal Subalgebra ◽  
Algebraically Closed Field ◽  
Cartan Type ◽  
Restricted Lie Algebra ◽  
Graded Lie Algebra

AbstractLetL=H(2r;n) be a graded Lie algebra of Hamiltonian type in the Cartan type series over an algebraically closed field of characteristicp>2. In the generalized restricted Lie algebra setup, any irreducible representation ofLcorresponds uniquely to a (generalized)p-characterχ. When the height ofχis no more than min {pni−pni−1∣i=1,2,…,2r}−2, the corresponding irreducible representations are proved to be induced from irreducible representations of the distinguished maximal subalgebraL0with the aid of an analogy of Skryabin’s category ℭ for the generalized Jacobson–Witt algebras and modulo finitely many exceptional cases. Since the exceptional simple modules have been classified, we can then give a full description of the irreducible representations withp-characters of height below this number.

Classification of irreducible representations of Lie algebra of vector fields on a torus

2016 ◽  
Vol 2016 (720) ◽  
Author(s):  
Yuly Billig ◽  
Vyacheslav Futorny
Keyword(s):  
Lie Algebra ◽  
Vector Fields ◽  
Irreducible Representations ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Standing Problem

AbstractWe solve a long standing problem of the classification of all simple modules with finite-dimensional weight spaces over Lie algebra of vector fields on

THE FOX PROBLEM FOR FREE RESTRICTED LIE ALGEBRAS

2008 ◽  
Vol 18 (02) ◽  
pp. 271-283 ◽  
Author(s):  
HAMID USEFI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Free Groups ◽  
Enveloping Algebra ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras

Let L be a free restricted Lie algebra and R a restricted ideal of L. Denote by u(L) the restricted enveloping algebra of L and by ω(L) the associative ideal of u(L) generated by L. The purpose of this paper is to identify the subalgebra R ∩ ωn(L)ω(R) in terms of R only. This problem is the analogue of the Fox problem for free groups.

The variety of nilpotent elements and invariant polynomial functions on the special algebra Sn

2015 ◽  
Vol 27 (3) ◽  
Author(s):  
Junyan Wei ◽  
Hao Chang ◽  
Xin Lu
Keyword(s):  
Lie Algebra ◽  
Invariant Polynomial ◽  
Polynomial Functions ◽  
Ground Field ◽  
Nilpotent Elements ◽  
Restricted Lie Algebra ◽  
Finite Dimensional ◽  
Special Algebra

AbstractIn the study of the variety of nilpotent elements in a Lie algebra, Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. In this paper, with the assumption that the ground field is algebraically closed of characteristic

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