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.

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 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.

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.

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.

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

Infinite Dimensional Boson-Fermion Representations of Lie Superalgebras D(M,N)

1984 ◽  
Vol 3 (4) ◽  
pp. 529-532 ◽  
Author(s):  
Han Qi-zhi ◽  
Liu Fu-sui ◽  
Sun Hong-zhou
Keyword(s):  
Lie Superalgebras ◽  
Infinite Dimensional
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