# 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} }

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…

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## 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

- Mathematics
- 1996

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

- Mathematics
- 1998

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

- Mathematics
- 1996

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…