BALINSKY–NOVIKOV SUPERALGEBRAS AND SOME INFINITE-DIMENSIONAL LIE SUPERALGEBRAS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250119 ◽  
Author(s):  
YUFENG PEI ◽  
CHENGMING BAI
Keyword(s):  
Direct Consequence ◽  
Lie Superalgebra ◽  
Explicit Construction ◽  
Lie Superalgebras ◽  
Bilinear Forms ◽  
Central Extensions ◽  
Infinite Dimensional

In this paper, we recall the Balinsky–Novikov (BN) superalgebras and revisit the approach of constructing an infinite-dimensional Lie superalgebra by a kind of affinization of a BN superalgebra. As an example, we give an explicit construction of Beltrami and Green–Schwarz–Witten (GSW) algebras from two isomorphic BN superalgebras, respectively, which proves that they are isomorphic as a direct consequence. Moreover, we consider the central extensions of the infinite-dimensional Lie superalgebras induced from BN superalgebras through certain bilinear forms on their corresponding BN superalgebras.

QUANTIZATION AND COCYCLES ON THE SUPERTORUS AND LARGE-N LIMITS FOR THE CLASSICAL LIE SUPERALGEBRAS

1991 ◽  
Vol 06 (03) ◽  
pp. 217-224 ◽  
Author(s):  
E.S. FRADKIN ◽  
V. Ya. LINETSKY
Keyword(s):  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Infinite Dimensional ◽  
Large N

The Poisson superbracket Lie superalgebra on the supertorus T2d|N is considered and its quantization is carried out. It is shown that there exists a non-trivial supercentral extension by means of 2d arbitrary c-numbers (when N is even), or 2d Grassmann numbers (when N is odd). It is shown that the infinite-dimensional superalgebras on the supertorus T2d|N can be considered as certain generalizations and large-M limits of the classical superalgebras A(M| M) and Q(M) (when N is even and odd respectively).

scholarly journals Bilinear forms and 2-dimensional cohomology

1986 ◽  
Vol 41 (2) ◽  
pp. 165-179
Author(s):  
Martin Moskowitz
Keyword(s):  
Cohomology Group ◽  
Nilpotent Lie Groups ◽  
Bilinear Forms ◽  
Locally Compact ◽  
Central Extensions ◽  
Direct Sums ◽  
Infinite Dimensional ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Cohomology Of Groups

AbstractThis paper calculates the central Borel 2 cocycles for certain 2-step nilpotent Lie groups G with values in the injectives A of the category of 2nd countable locally compact abelian groups. The G's include, among others, all groups locally isomorphic to a Heisenberg group. The A's are direct sums of vector groups and (possibly infinite dimensional) tori, and in particular include R, T, and Cx. The main results are as follows.(4.1) Every symmetric central 2 cocycle is trivial.(4.2) Every central 2 cocycle is cohomologous with a skew symmetric bimultiplicative one (which is necessarily jointly continuous by [7]).(4.3) The corresponding cohomology group H2cent (G, A) is calculated as the skew symmetric jointly continuous bimultiplicative maps modulo Homcont ([G, G]–, A).These results generalize the case when G is a connected abelian Lie group and A = T, due to Kleppner [3]. Using standard facts of the cohomology of groups they can be interpreted as classifying all continuous central extensions (1) → A → E → G → (1) of the group G by the abelian group A. Finally some counterexamples are given to extending these results.

scholarly journals The classification of modular Lie superalgebras of type M

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Lili Ma ◽  
Liangyun Chen
Keyword(s):  
Intrinsic Property ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Characteristic P ◽  
Infinite Dimensional ◽  
Modular Lie Superalgebra ◽  
Nilpotent Elements ◽  
Natural Filtration ◽  
Modular Lie Superalgebras

AbstractThe natural filtration of the infinite-dimensional simple modular Lie superalgebra M over a field of characteristic p > 2 is proved to be invariant under automorphisms by discussing ad-nilpotent elements. Moreover, an intrinsic property is obtained and all the infinite-dimensional simple modular Lie superalgebras M are classified up to isomorphisms. As an application, a property of automorphisms of M is given.

THE MODIFIED CLASSICAL YANG-BAXTER EQUATION AND SUPERSYMMETRIC GEL’FAND-DIKII BRACKETS

1993 ◽  
Vol 08 (02) ◽  
pp. 129-137 ◽  
Author(s):  
C.M. YUNG
Keyword(s):  
Pseudodifferential Operators ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Baxter Equation ◽  
Poisson Brackets ◽  
Trace Form ◽  
Kp Hierarchy ◽  
Infinite Dimensional

The classical Yang-Baxter equation as formulated by Semenov-Tyan-Shanskii is generalized to the case of Lie superalgebras [Formula: see text], for Grassmann even Yang-Baxter operators ℛ. When ℛ is “unitary” with respect to a super trace form defined on [Formula: see text], we prove the existence of two natural Poisson brackets on the dual [Formula: see text]*. If [Formula: see text] is the infinite-dimensional Lie superalgebra of N=1 super pseudodifferential operators, we recover the super Gel’fand-Dikii brackets underlying the N=1 super KP hierarchy and its reductions.

scholarly journals Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields

2017 ◽  
Vol 15 (1) ◽  
pp. 1332-1343
Author(s):  
Liping Sun ◽  
Wende Liu
Keyword(s):  
Vector Fields ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Infinite Dimensional

Abstract According to the classification by Kac, there are eight Cartan series and five exceptional Lie superalgebras in infinite-dimensional simple linearly compact Lie superalgebras of vector fields. In this paper, the Hom-Lie superalgebra structures on the five exceptional Lie superalgebras of vector fields are studied. By making use of the ℤ-grading structures and the transitivity, we prove that there is only the trivial Hom-Lie superalgebra structures on exceptional simple Lie superalgebras. This is achieved by studying the Hom-Lie superalgebra structures only on their 0-th and (−1)-th ℤ-components.

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.

scholarly journals REPRESENTATIONS OF THE EXCEPTIONAL LIE SUPERALGEBRA E(3,6) III: CLASSIFICATION OF SINGULAR VECTORS

2005 ◽  
Vol 04 (01) ◽  
pp. 15-57 ◽  
Author(s):  
VICTOR G. KAC ◽  
ALEXEI RUDAKOV
Keyword(s):  
Vector Fields ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Irreducible Representations ◽  
Verma Modules ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Singular Vectors ◽  
Zero Degree

We continue the study of irreducible representations of the exceptional Lie superalgebra E(3,6). This is one of the two simple infinite-dimensional Lie superalgebras of vector fields which have a Lie algebra sℓ(3) × sℓ(2) × gℓ(1) as the zero degree component of its consistent ℤ-grading. We provide the classification of the singular vectors in the degenerate Verma modules over E(3,6), completing thereby the classification and construction of all irreducible E(3,6)-modules that are L0-locally finite.

Invariant bilinear forms on Leibniz superalgebras

2021 ◽  
pp. 2250098
Author(s):  
Saïd Benayadi ◽  
Fahmi Mhamdi
Keyword(s):  
Bilinear Form ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Bilinear Forms ◽  
Algebraic Structures ◽  
New Type ◽  
The Right ◽  
Appl Math

In [S. Benayadi, F. Mhamdi and S. Omri, Quadratic (resp. symmetric) Leibniz superalgebras, Commun. Algebra, https://doi.org/10.1080/00927872.2020.1850751 ], the quadratic Leibniz superalgebras, which are (left or right) Leibniz superalgebras provided with even supersymmetric non-degenerate and associative bilinear forms, were investigated. Besides the notion of associativity, there are other kinds of invariance for bilinear forms on Leibniz superalgebras, namely the left invariance and the right invariance. In this paper, we investigate the Leibniz superalgebras endowed with even, supersymmetric, non-degenerate and left invariant bilinear forms and highlight the links between these superalgebras and some other algebraic structures. More precisely, every symmetric Leibniz superalgebra provided with such a bilinear form gives rise to a new type of superalgebra which we call LS-Lie superalgebra. We study LS-Lie superalgebras and we give some interesting informations on the structure of these superalgebras by using certain extensions introduced in [S. Benayadi and F. Mhamdi, Odd-quadratic Leibniz superalgebras, Adv. Pure Appl. Math. 10(4) (2019) 287–298; S. Benayadi, F. Mhamdi and S. Omri, Quadratic (resp. symmetric) Leibniz superalgebras, Commun. Algebra, https://doi.org/10.1080/00927872.2020.1850751 ]. Further, several nontrivial examples of LS-Lie superalgebras are included. Finally, we give similar results for Leibniz superalgebras with right invariant bilinear forms.

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