Tensor fields of mixed Young symmetry type and N-complexes

Details Tensor fields of mixed Young symmetry type and N-complexes Tensor fields of mixed Young symmetry type and N-complexes

2001-10-09

arxiv.org/abs/math/0110088v1

Commun.Math.Phys. 226 (2002) 393-418

Mathematics

Quantum Algebra

47 pages

Scientific paper

10.1007/s002200200610

We construct $N$-complexes of non completely antisymmetric irreducible tensor fields on $\mathbb R^D$ which generalize the usual complex $(N=2)$ of differential forms. Although, for $N\geq 3$, the generalized cohomology of these $N$-complexes is non trivial, we prove a generalization of the Poincar\'e 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 Poincar\'e lemma. We furthermore identify the nontrivial part of the generalized cohomology. Many of the results presented here were announced in [10].

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