A Class of Finite-dimensional Lie Superalgebras of Hamiltonian Type

2011 ◽  
Vol 18 (02) ◽  
pp. 347-360 ◽  
Author(s):  
Li Ren ◽  
Qiang Mu ◽  
Yongzheng Zhang
Keyword(s):  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Prime Characteristic ◽  
Cartan Type ◽  
Finite Dimensional ◽  
Derivation Superalgebra

A class of finite-dimensional Cartan-type Lie superalgebras H(n,m) over a field of prime characteristic is studied in this paper. We first determine the derivation superalgebra of H(n,m). Then we obtain that H(n,m) is restrictable and it is an extension of the Lie superalgebra [Formula: see text]. Finally, we prove that H(n,m) is isomorphic to a subalgebra of the restricted Hamiltonian Lie superalgebra [Formula: see text].

The Finite-dimensional Modular Lie Superalgebra Γ

2010 ◽  
Vol 17 (03) ◽  
pp. 525-540 ◽  
Author(s):  
Xiaoning Xu ◽  
Yongzheng Zhang ◽  
Liangyun Chen
Keyword(s):  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Explicit Description ◽  
Cartan Type ◽  
Modular Lie Superalgebra ◽  
New Family ◽  
Finite Dimensional ◽  
Derivation Superalgebra ◽  
Modular Lie Superalgebras

A new family of finite-dimensional modular Lie superalgebras Γ is defined. The simplicity and generators of Γ are studied and an explicit description of the derivation superalgebra of Γ is given. Moreover, it is proved that Γ is not isomorphic to any known Lie superalgebra of Cartan type.

scholarly journals The Natural Filtration of Finite Dimensional Modular Lie Superalgebras of Special Type

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Keli Zheng ◽  
Yongzheng Zhang
Keyword(s):  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Prime Characteristic ◽  
Modular Lie Superalgebra ◽  
Finite Dimensional ◽  
Natural Filtration ◽  
Modular Lie Superalgebras

This paper is concerned with the natural filtration of Lie superalgebraS(n,m)of special type over a field of prime characteristic. We first construct the modular Lie superalgebraS(n,m). Then we prove that the natural filtration ofS(n,m)is invariant under its automorphisms.

FINITE-DIMENSIONAL SPECIAL ODD HAMILTONIAN SUPERALGEBRAS IN PRIME CHARACTERISTIC

2009 ◽  
Vol 11 (04) ◽  
pp. 523-546 ◽  
Author(s):  
WENDE LIU ◽  
YINGHUA HE
Keyword(s):  
Lie Superalgebras ◽  
The Other ◽  
Prime Characteristic ◽  
Characteristic P ◽  
Cartan Type ◽  
New Family ◽  
Finite Dimensional ◽  
Modular Lie Superalgebras

In this paper, we study a new family of finite-dimensional simple Lie superalgebras of Cartan type over a field of characteristic p > 3, the so-called special odd Hamiltonian superalgebras. The spanning sets are first given and then the grading structures are described explicitly. Finally, the simplicity and the dimension formulas are determined. As application, using the dimension formulas, we make a comparison between the special odd Hamiltonian superalgebras and the other known families of finite-dimensional simple modular Lie superalgebras of Cartan type.

Automorphism groups of modular graded Lie superalgebras of Cartan-type

2017 ◽  
Vol 16 (03) ◽  
pp. 1750050
Author(s):  
Wende Liu ◽  
Jixia Yuan
Keyword(s):  
Automorphism Groups ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Infinite Dimensional ◽  
Cartan Type ◽  
Type Formula ◽  
Finite Dimensional ◽  
Normal Series

Suppose the underlying field is of characteristic [Formula: see text]. In this paper, we prove that the automorphisms of the finite-dimensional graded (non-restircited) Lie superalgebras of Cartan-type [Formula: see text] [Formula: see text] [Formula: see text] and [Formula: see text] can uniquely extend to the ones of the infinite-dimensional Lie superalgebra of Cartan-type [Formula: see text]. Then a concrete group embedding from [Formula: see text] into [Formula: see text] is established, where [Formula: see text] is any finite-dimensional Lie superalgebra of Cartan-type [Formula: see text] or [Formula: see text] and [Formula: see text] is the underlying (associative) superalgebra of [Formula: see text]. The normal series of the automorphism groups of [Formula: see text] are also considered.

Automorphisms and p-Characters of Restricted Lie Superalgebras of Cartan Type

2011 ◽  
Vol 18 (03) ◽  
pp. 397-410 ◽  
Author(s):  
Jixia Yuan ◽  
Yan Chen ◽  
Wende Liu
Keyword(s):  
Automorphism Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Irreducible Representations ◽  
Prime Characteristic ◽  
Cartan Type ◽  
Restricted Lie Superalgebra ◽  
Normal Series ◽  
Standard Normal

Let X be a restricted Lie superalgebra of Cartan type W, S, H or K over a field of prime characteristic. In this paper, we describe the quotients of the standard normal series of the automorphism group of X. As an application, the results above are used to discuss the p-characters of the irreducible representations for X.

scholarly journals On Low-Dimensional Complex ω-Lie Superalgebras

2021 ◽  
Author(s):  
Jia Zhou ◽  
Liangyun Chen
Keyword(s):  
Automorphism Group ◽  
Representation Theory ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
3 Dimensional ◽  
Finite Dimensional ◽  
Low Dimensional ◽  
Derivation Superalgebra

Abstract Let (g, [−, −], ω) be a finite-dimensional complex ω-Lie superalgebra. In this paper, we introduce the notions of derivation superalgebra Der(g) and the automorphism group Aut(g) of (g, [−, −], ω). We study Derω (g) and Autω (g), which are superalgebra of Der(g) and subgroup of Aut(g), respectively. For any 3-dimensional or 4-dimensional complex ω-Lie superalgebra g, we explicitly calculate Der(g) and Aut(g), and obtain Jordan standard forms of elements in the two sets. We also study representation theory of ω-Lie superalgebras and give a conclusion that all nontrivial non-ω-Lie 3-dimensional and 4-dimensional ω-Lie superalgebras are multiplicative, as well as we show that any irreducible respresentation of the 4-dimensional ω-Lie superalgebra P2,k(k 6= 0, −1) is 1-dimensional.

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.

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 Equivariant Map Queer Lie Superalgebras

2016 ◽  
Vol 68 (2) ◽  
pp. 258-279 ◽  
Author(s):  
Lucas Calixto ◽  
Adriano Moura ◽  
Alistair Savage
Keyword(s):  
Finite Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Equivariant Map ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Regular Maps ◽  
A Finite Group ◽  
Special Case

AbstractAn equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a finite group Γ acting on X and q. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant finitely supported maps from X to the set of isomorphism classes of irreducible finite-dimensional representations of q. In the special case where X is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

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