The q-Analog Klein Bottle Lie Algebra

2014 ◽  
Vol 21 (04) ◽  
pp. 561-574
Author(s):  
Cuipo Jiang ◽  
Jingjing Jiang ◽  
Yufeng Pei
Keyword(s):  
Lie Algebra ◽  
Bilinear Form ◽  
Central Extension ◽  
Klein Bottle ◽  
Universal Central Extension ◽  
Finitely Generated ◽  
Invariant Bilinear Form ◽  
Infinite Dimensional ◽  
Simple Lie Algebra ◽  
Infinite Dimensional Lie Algebra

In this paper, we study an infinite-dimensional Lie algebra ℬq, called the q-analog Klein bottle Lie algebra. We show that ℬq is a finitely generated simple Lie algebra with a unique (up to scalars) symmetric invariant bilinear form. The derivation algebra and the universal central extension of ℬq are also determined.

HIROTA BILINEAR FORM FOR THE SUPER-KdV HIERARCHY

1993 ◽  
Vol 08 (18) ◽  
pp. 1739-1745 ◽  
Author(s):  
I.N. MCARTHUR ◽  
C.M. YUNG
Keyword(s):  
Lie Algebra ◽  
Bilinear Form ◽  
Infinite Dimensional ◽  
Kdv Hierarchy ◽  
Hirota Bilinear Form ◽  
Group Orbit ◽  
Orbit Equation ◽  
Infinite Dimensional Lie Algebra ◽  
Hirota Bilinear

The integrability of the super-KdV hierarchy suggests that it can be written in Hirota bilinear form as the group orbit equation for some infinite-dimensional Lie algebra. We show how the first few equations in the hierarchy can be written in Hirota bilinear form. We also conjecture a bilinear expression for the whole super-KdV hierarchy and check it to reasonably high orders. A by-product is an expression for the ordinary KdV hierarchy which provides an alternative to the ones obtained by Date et al. and Kac and Wakimoto.

scholarly journals Weyl modules and Weyl functors for hyper-map algebras

2020 ◽  
pp. 2140007
Author(s):  
Angelo Bianchi ◽  
Samuel Chamberlin
Keyword(s):  
Lie Algebra ◽  
Finitely Generated ◽  
Weyl Modules ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Universal Properties ◽  
Definition Of ◽  
Unitary Algebra

We investigate the representations of the hyperalgebras associated to the map algebras [Formula: see text], where [Formula: see text] is any finite-dimensional complex simple Lie algebra and [Formula: see text] is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.

Infinite-Dimensional Lie Algebra

2004 ◽  
pp. 202-202
Author(s):  
John Howie ◽  
Steven Duplij ◽  
Ali Mostafazadeh ◽  
Masaki Yasue ◽  
Vladimir Ivashchuk ◽  
...  
Keyword(s):  
Lie Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Lie Algebra

scholarly journals A COHOMOLOGICAL INTERPRETATION OF THE MIGDAL–MAKEENKO EQUATIONS

2002 ◽  
Vol 17 (08) ◽  
pp. 481-489 ◽  
Author(s):  
A. AGARWAL ◽  
S. G. RAJEEV
Keyword(s):  
Lie Algebra ◽  
Equations Of Motion ◽  
Generating Functional ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Change Of Variables ◽  
Anomalous Term ◽  
Large N ◽  
Infinite Dimensional Lie Algebra ◽  
Cohomological Interpretation

The equations of motion of quantum Yang–Mills theory (in the planar "large-N" limit), when formulated in loop-space are shown to have an anomalous term, which makes them analogous to the equations of motion of WZW models. The anomaly is the Jacobian of the change of variables from the usual ones, i.e. the connection one-form A, to the holonomy U. An infinite-dimensional Lie algebra related to this change of variables (the Lie algebra of loop substitutions) is developed, and the anomaly is interpreted as an element of the first cohomology of this Lie algebra. The Migdal–Makeenko equations are shown to be the condition for the invariance of the Yang–Mills generating functional Z under the action of the generators of this Lie algebra. Connections of this formalism to the collective field approach of Jevicki and Sakita are also discussed.

scholarly journals Matrix 3-Lie superalgebras and BRST supersymmetry

2017 ◽  
Vol 14 (11) ◽  
pp. 1750160 ◽  
Author(s):  
Viktor Abramov
Keyword(s):  
Clifford Algebra ◽  
Lie Algebra ◽  
Vector Fields ◽  
Jacobi Identity ◽  
Lie Superalgebras ◽  
Infinite Dimensional ◽  
Algebra Approach ◽  
The Matrix ◽  
Infinite Dimensional Lie Algebra

Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper, we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras [Formula: see text] if instead of the trace of a matrix we make use of the supertrace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the supertrace satisfies a graded ternary Filippov–Jacobi identity. In two particular cases of [Formula: see text] and [Formula: see text], we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

scholarly journals SYMMETRIES AND LIE ALGEBRA OF THE DIFFERENTIAL–DIFFERENCE KADOMSTEV–PETVIASHVILI HIERARCHY

2010 ◽  
Vol 24 (10) ◽  
pp. 1033-1042 ◽  
Author(s):  
XIAN-LONG SUN ◽  
DA-JUN ZHANG ◽  
XIAO-YING ZHU ◽  
DENG-YUAN CHEN
Keyword(s):  
Lie Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Lie Algebra

By introducing suitable non-isospectral flows, we construct two sets of symmetries for the isospectral differential–difference Kadomstev–Petviashvili hierarchy. The symmetries form an infinite dimensional Lie algebra.

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