The Even Part of Finite-Dimensional Modular Lie Superalgebra Γ

2021 ◽  
Vol 28 (03) ◽  
pp. 479-496
Author(s):  
Yusi Fan ◽  
Xiaoning Xu ◽  
Liangyun Chen
Keyword(s):  
Lie Algebra ◽  
Lie Superalgebra ◽  
Base Field ◽  
Modular Lie Superalgebra ◽  
Derivation Algebra ◽  
Finite Dimensional ◽  
Generator Sets

Let [Formula: see text] be the underlying base field of characteristic [Formula: see text] and denote by [Formula: see text] the even part of the finite-dimensional Lie superalgebra [Formula: see text]. We give the generator sets of the Lie algebra [Formula: see text]. Using certain properties of the canonical tori of [Formula: see text], we describe explicitly the derivation algebra of [Formula: see text].

DERIVATIONS OF THE EVEN PARTS FOR MODULAR LIE SUPERALGEBRAS OF CARTAN TYPE W AND S

2007 ◽  
Vol 17 (04) ◽  
pp. 661-714 ◽  
Author(s):  
WENDE LIU ◽  
YONGZHENG ZHANG
Keyword(s):  
Lie Algebras ◽  
Lie Superalgebra ◽  
Adjoint Representation ◽  
Lie Superalgebras ◽  
Base Field ◽  
Characteristic P ◽  
Derivation Algebra ◽  
Finite Dimensional ◽  
Outer Derivation ◽  
Generator Sets

Let 𝔽 be the underlying base field of characteristic p < 3 and denote by [Formula: see text] and [Formula: see text] the even parts of the finite-dimensional generalized Witt Lie superalgebra W and the special Lie superalgebra S, respectively. We first give the generator sets of the Lie algebras [Formula: see text] and [Formula: see text]. Using certain properties of the canonical tori of [Formula: see text] and [Formula: see text], we then determine the derivation algebra of [Formula: see text] and the derivation space of [Formula: see text] to [Formula: see text], where [Formula: see text] is viewed as a [Formula: see text]-module by means of the adjoint representation. As a result, we describe explicitly the derivation algebra of [Formula: see text]. Furthermore, we prove that the outer derivation algebras of [Formula: see text] and [Formula: see text] are abelian Lie algebras or metabelian Lie algebras with explicit structure. In particular, we give the dimension formulas of the derivation algebras and outer derivation algebras of [Formula: see text] and [Formula: see text]. Thus, we may make a comparison between the even parts of the (outer) superderivation algebras of W and S and the (outer) derivation algebras of the even parts of W and S, respectively.

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.

Double Derivations of n-Lie Superalgebras

2018 ◽  
Vol 25 (01) ◽  
pp. 161-180
Author(s):  
Bing Sun ◽  
Liangyun Chen ◽  
Xin Zhou
Keyword(s):  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
General Linear ◽  
Base Field ◽  
Inner Derivation ◽  
Derivation Algebra ◽  
General Linear Lie Superalgebra

Let 𝔤 be an n-Lie superalgebra. We study the double derivation algebra [Formula: see text] and describe the relation between [Formula: see text] and the usual derivation Lie superalgebra Der(𝔤). We show that the set [Formula: see text] of all double derivations is a subalgebra of the general linear Lie superalgebra gl(𝔤) and the inner derivation algebra ad(𝔤) is an ideal of [Formula: see text]. We also show that if 𝔤 is a perfect n-Lie superalgebra with certain constraints on the base field, then the centralizer of ad(𝔤) in [Formula: see text] is trivial. Finally, we give that for every perfect n-Lie superalgebra 𝔤, the triple derivations of the derivation algebra Der(𝔤) are exactly the derivations of Der(𝔤).

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.

Second Cohomology of the Modular Lie Superalgebra of Cartan Type K

2009 ◽  
Vol 16 (02) ◽  
pp. 309-324 ◽  
Author(s):  
Wenjuan Xie ◽  
Yongzheng Zhang
Keyword(s):  
Cohomology Group ◽  
Lie Superalgebra ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Cartan Type ◽  
Modular Lie Superalgebra ◽  
Finite Dimensional ◽  
Type K

Let 𝔽 be an algebraically closed field and char 𝔽 = p > 3. In this paper, we determine the second cohomology group of the finite-dimensional Contact superalgebra K(m,n,t).

scholarly journals MAD SUBALGEBRAS AND LIE SUBALGEBRAS OF AN ENVELOPING ALGEBRA

2010 ◽  
Vol 82 (3) ◽  
pp. 401-423
Author(s):  
XIN TANG
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Two Dimensional ◽  
Base Field ◽  
Inner Derivation ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Characterization Theorems

AbstractLet 𝒰(𝔯(1)) denote the enveloping algebra of the two-dimensional nonabelian Lie algebra 𝔯(1) over a base field 𝕂. We study the maximal abelian ad-nilpotent (mad) associative subalgebras and finite-dimensional Lie subalgebras of 𝒰(𝔯(1)). We first prove that the set of noncentral elements of 𝒰(𝔯(1)) admits the Dixmier partition, 𝒰(𝔯(1))−𝕂=⋃ 5i=1Δi, and establish characterization theorems for elements in Δi, i=1,3,4. Then we determine the elements in Δi, i=1,3 , and describe the eigenvalues for the inner derivation ad Bx,x∈Δi, i=3,4 . We also derive other useful results for elements in Δi, i=2,3,4,5 . As an application, we find all framed mad subalgebras of 𝒰(𝔯(1)) and determine all finite-dimensional nonabelian Lie algebras that can be realized as Lie subalgebras of 𝒰(𝔯(1)) . We also study the realizations of the Lie algebra 𝔯(1) in 𝒰(𝔯(1)) in detail.

Finite-dimensional simple modular Lie superalgebra ℳ

2012 ◽  
Vol 8 (2) ◽  
pp. 411-441 ◽  
Author(s):  
Lili Ma ◽  
Liangyun Chen ◽  
Yongzheng Zhang
Keyword(s):  
Lie Superalgebra ◽  
Modular Lie Superalgebra ◽  
Finite Dimensional

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.

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