scholarly journals The natural operators similar to the twisted courant bracket on couples of vector fields and p-forms

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4071-4078
Author(s):  
Włodzimierz Mikulski
Keyword(s):  
Vector Fields ◽  
Jacobi Identity ◽  
Natural Numbers ◽  
Normalization Condition ◽  
Courant Bracket ◽  
Natural Operator ◽  
Bilinear Operators ◽  
Natural Operators

Given natural numbers m and p with m ? p + 2 ? 3, all Mfm-natural operators A sending closed (p+2)-forms H on m-manifolds M into R-bilinear operators AH transforming pairs of couples of vector fields and p-forms on M into couples of vector fields and p-forms on M are found. If m ? p + 2 ? 3, all Mfm-natural operators A (as above) such that AH satisfies the Jacobi identity in Leibniz form are extracted, and that the twisted Courant bracket [-,-]H is the unique Mfm-natural operator AH (as above) satisfying the Jacobi identity in Leibniz form and some normalization condition is deduced.

scholarly journals On the twisted Dorfman-Courant like brackets

2020 ◽  
Vol 40 (6) ◽  
pp. 703-723
Author(s):  
Włodzimierz M. Mikulski
Keyword(s):  
Vector Bundle ◽  
Jacobi Identity ◽  
Courant Bracket ◽  
Bilinear Operators ◽  
Restricted Condition ◽  
Linear Sections ◽  
Natural Operators

There are completely described all \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) which, like to the Dorfman-Courant bracket, send closed linear \(3\)-forms \(H\in\Gamma^{l-\rm{clos}}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^{\infty}\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE\oplus T^*E)\times \Gamma^l_E(TE\oplus T^*E)\to \Gamma^l_E(TE\oplus T^*E)\] transforming pairs of linear sections of \(TE\oplus T^*E\to E\) into linear sections of \(TE\oplus T^*E\to E\). Then all such \(C\) which also, like to the twisted Dorfman-Courant bracket, satisfy both some "restricted" condition and the Jacobi identity in Leibniz form are extracted.

scholarly journals On the gauge-natural operators similar to the twisted Dorfman-Courant bracket

2021 ◽  
Vol 41 (2) ◽  
pp. 205-226
Author(s):  
Włodzimierz M. Mikulski
Keyword(s):  
Vector Bundle ◽  
Jacobi Identity ◽  
Courant Bracket ◽  
Bilinear Operators ◽  
Linear Sections ◽  
Natural Operators

All \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) sending linear \(3\)-forms \(H \in \Gamma^{l}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^\infty\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE \oplus T^*E)\times \Gamma^l_E(TE \oplus T^*E)\to \Gamma^l_E(TE \oplus T^*E)\] transforming pairs of linear sections of \(TE \oplus T^*E \to E\) into linear sections of \( TE \oplus T^*E \to E\) are completely described. The complete descriptions is given of all generalized twisted Dorfman-Courant brackets \(C\) (i.e. \(C\) as above such that \(C_0\) is the Dorfman-Courant bracket) satisfying the Jacobi identity for closed linear \(3\)-forms \(H\). An interesting natural characterization of the (usual) twisted Dorfman-Courant bracket is presented.

scholarly journals On the Courant bracket on couples of vector fields and \(p\)-forms

2018 ◽  
Vol 72 (2) ◽  
pp. 29
Author(s):  
Miroslav Doupovec ◽  
Jan Kurek ◽  
Włodzimierz Mikulski
Keyword(s):  
Vector Fields ◽  
Bilinear Operator ◽  
Courant Bracket ◽  
Bilinear Operators

If \(m\geq p+1\geq 2\) (or \(m=p\geq 3\)), all  natural bilinear  operators \(A\) transforming pairs of couples of vector fields and \(p\)-forms on \(m\)-manifolds \(M\) into couples of vector fields and \(p\)-forms on \(M\) are described. It is observed that  any natural skew-symmetric bilinear operator \(A\) as above coincides with the generalized Courant bracket up to three (two, respectively) real constants.

ON SOME NATURAL OPERATORS IN VECTOR FIELDS

2003 ◽  
Vol 36 (1) ◽  
Author(s):  
Włodzimierz M. Mikulski
Keyword(s):  
Vector Fields ◽  
Natural Operators

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 CONSISTENT INTERACTIONS BETWEEN GAUGE FIELDS AND LOCAL BRST COHOMOLOGY: THE EXAMPLE OF YANG-MILLS MODELS

1994 ◽  
Vol 03 (01) ◽  
pp. 139-144 ◽  
Author(s):  
G. BARNICH ◽  
M. HENNEAUX ◽  
R. TATAR
Keyword(s):  
Vector Fields ◽  
Jacobi Identity ◽  
Second Order ◽  
Gauge Fields ◽  
Coupling Constant ◽  
Yang Mills ◽  
First Order ◽  
Gauge Symmetries ◽  
Brst Cohomology ◽  
Structure Constants

Recent results on the cohomological reformulation of the problem of consistent interactions between gauge fields are illustrated in the case of the Yang-Mills models. By evaluating the local BRST cohomology through descent equation techniques, it is shown (i) that there is a unique local, Poincaré invariant cubic vertex for free gauge vector fields which preserves the number of gauge symmetries to first order in the coupling constant; and (ii) that consistency to second order in the coupling constant requires the structure constants appearing in the cubic vertex to fulfill the Jacobi identity. The known uniqueness of the Yang-Mills coupling is therefore rederived through cohomological arguments.

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