# Tensor Fields of Mixed Young Symmetry Type¶and N-Complexes

```@article{DuboisViolette2002TensorFO,
title={Tensor Fields of Mixed Young Symmetry Type¶and N-Complexes},
author={Michel Dubois-Violette and M Henneaux},
journal={Communications in Mathematical Physics},
year={2002},
volume={226},
pages={393-418}
}```
• Published 9 October 2001
• Mathematics
• Communications in Mathematical Physics
Abstract: We construct N-complexes of non-completely antisymmetric irreducible tensor fields on ℝD which generalize the usual complex (N=2) of differential forms. Although, for N≥ 3, the generalized cohomology of these N-complexes is nontrivial, we prove a generalization of the Poincaré lemma. To that end we use a technique reminiscent of the Green ansatz for parastatistics. Several results which appeared in various contexts are shown to be particular cases of this generalized Poincaré lemma…
The graded differential geometry of mixed symmetry tensors
• Mathematics
Archivum Mathematicum
• 2019
We show how the theory of \$\mathbb{Z}_2^n\$ -manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual
Tensor Product of N-complexes 2 Preliminary : From the Tensor Product of Complexes to Graded Differential Algebras
It is known that the notion of graded differential algebra coincides with the notion of monoid in the monoidal category of complexes. By using the monoidal structure introduced by M. Kapranov for the
Tensor Product of N-complexes and Generalization of Graded Differential Algebras
It is known that the notion of graded differential algebra coincides with the notion of monoid in the monoidal category of complexes. By using the monoidal structure introduced by M.Kapranov forthe
Consistent deformations of [p,p]-type gauge field theories
• Physics
• 2004
Using BRST-cohomological techniques, we analyze the consistent deformations of theories describing free tensor gauge fields whose symmetries are represented by Young tableaux made of two columns of
U(N) spinning particles and higher spin equations on complex manifolds
• Mathematics
• 2009
Guided by a spinning particle model with U(N)-extended supergravity on the worldline we derive higher spin equations on complex manifolds. Their minimal formulation is in term of gauge fields which
Consistent deformations of dual formulations of linearized gravity: A No go result
• Mathematics
• 2003
The consistent, local, smooth deformations of the dual formulation of linearized gravity involving a tensor field in the exotic representation of the Lorentz group with Young symmetry type (D-3,1)
N-Complexes and Higher Spin Gauge Fields
N-complexes have been argued recently to be algebraic structures relevant to the description of higher spin gauge fields. N-complexes involve a linear operator d that fulfills dN = 0 and that defines
Functional analytic issues in Z_2 ^n Geometry
• Mathematics
• 2018
We show that the function sheaf of a Z2 -manifold is a nuclear Fréchet sheaf of Z2 graded Z2 -commutative associative unital algebras. Further, we prove that the components of the pullback sheaf
Massless spin two field S duality
• Mathematics
• 2002
We present a review of the homological algebra tools involved in the standard de Rham theory and their subsequent generalizations relevant for the understanding of free massless higher spin gauge

## References

SHOWING 1-10 OF 25 REFERENCES
Generalized Cohomology for Irreducible Tensor Fields of Mixed Young Symmetry Type
• Mathematics
• 1999
We construct N-complexes of noncompletely antisymmetric irreducible tensor fields on ℝD, thereby generalizing the usual complex (N=2) of differential forms. These complexes arise naturally in the
On the q-analog of homological algebra
This is an attempt to generalize some basic facts of homological algebra to the case of "complexes" in which the differential satisfies the condition \$d^N=0\$ instead of the usual \$d^2=0\$. Instead of
Generalized Homologies for the Zero Modes of the SU(2) WZNW Model
• Mathematics
• 1999
We generalize the BRS method for the (finite-dimensional) quantum gauge theory involved in the zero modes of the monodromy extended SU(2) WZNW model. The generalization consists of a nilpotent
\$d^N=0\$ : Generalized homology
We study the generalized homology associated with a nilpotent endomorphism \$d\$ satisfying \$d^N=0\$. For simplicial modules we construct such nilpotent endomorphisms and we prove a general result
Generalized differential spaces withdN=0 and theq-differential calculus
We present some results concerning the generalized homologies associated with nilpotent endomorphismsd such thatdN=0 for some integerN≥2. We then introduce the notion of gradedq-differential algebra
Algèbre Homologique des N-Complexes et Homologie de Hochschild aux Racines de l'Unité
• Mathematics
• 1998
We set up a homological algebra for Af-complexes, which are graded modules together with a degree —1 endomorphism d satisfying dN = 0. We define Tor- and Ext-groups for Ar-complexes and we compute
Systematics of Higher Spin Gauge Fields
• Physics
• 1980
Free-field theories for symmetric tensor and tensor-spinor gauge fields have recently been obtained which describe massless particles of arbitrary integer or half-integer spin. An independent
Universal \$q\$-differential calculus and \$q\$-analog of homological algebra
• Mathematics
• 1996
We recall the definition of \$q\$-differential algebras and discuss some representative examples. In particular we construct the \$q\$-analog of the Hochschild coboundary. We then construct the universal