Structure of some ℤ-graded lie superalgebras of vector fields

1999 ◽  
Vol 4 (2-3) ◽  
pp. 219-272 ◽  
Author(s):  
S. J. Cheng ◽  
V. Kac
Keyword(s):  
Vector Fields ◽  
Lie Superalgebras

scholarly journals Five exceptional simple Lie superalgebras of vector fields

1999 ◽  
Vol 33 (3) ◽  
pp. 208-219 ◽  
Author(s):  
I. M. Shchepochkina
Keyword(s):  
Vector Fields ◽  
Lie Superalgebras

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 Errata to "Structure of Some ℤ-graded Lie Superalgebras of Vector Fields''

2004 ◽  
Vol 9 (4) ◽  
pp. 399-400 ◽  
Author(s):  
N. Cantarini ◽  
S.-J. Cheng ◽  
V. Kac
Keyword(s):  
Vector Fields ◽  
Lie Superalgebras

Lie superalgebras of vector fields on flag supermanifolds

2008 ◽  
Vol 63 (2) ◽  
pp. 394-396 ◽  
Author(s):  
E G Vishnyakova
Keyword(s):  
Vector Fields ◽  
Lie Superalgebras

scholarly journals COMPUTATION OF COHOMOLOGY OF LIE SUPERALGEBRAS OF VECTOR FIELDS

2000 ◽  
Vol 11 (02) ◽  
pp. 397-413 ◽  
Author(s):  
V. V. KORNYAK
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Mathematical Physics ◽  
Lie Superalgebras ◽  
Graded Algebras ◽  
Topological Information ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Cohomology Of Lie Algebras ◽  
Mathematics And Physics

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. It carries most fundamental ("topological") information about algebra under consideration. At present, because of the need for very tedious algebraic computation, the explicitly computed cohomology for different classes of Lie (super)algebras is known only in a few cases. That is why application of computer algebra methods is important for this problem. We describe here an algorithm and its C implementation for computing the cohomology of Lie algebras and superalgebras. The program can proceed finite-dimensional algebras and infinite-dimensional graded algebras with finite-dimensional homogeneous components. Among the last algebras, Lie algebras and superalgebras of formal vector fields are most important. We present some results of computation of cohomology for Lie superalgebras of Buttin vector fields and related algebras. These algebras being super-analogs of Poisson and Hamiltonian algebras have found many applications to modern supersymmetric models of theoretical and mathematical physics.

scholarly journals Explicit Bracket in an Exceptional Simple Lie Superalgebra

1998 ◽  
Vol 08 (04) ◽  
pp. 479-495 ◽  
Author(s):  
Irina Shchepochkina ◽  
Gerhard Post
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Vector Fields ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Exact Form ◽  
3 Dimensional

This note is devoted to a more detailed description of one of the five simple exceptional Lie superalgebras of vector fields, [Formula: see text] a subalgebra of [Formula: see text]. We derive differential equations for its elements, and solve these equations. Hence we get an exact form for the elements of [Formula: see text]. Moreover we realize [Formula: see text] by "glued" pairs of generating functions on a (3∣3)-dimensional periplectic (odd symplectic) supermanifold and describe the bracket explicitly.

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.

Sign in / Sign up

Export Citation Format

Share Document