September 93

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Journal of Geometry and Physics 15 (1995) 333-352

**Abstract :**
We define currents on a Grassmann algebra *Gr*(*N*) with *N* generators as distributions on its exterior algebra (using the
symmetric wedge product). We interpret the currents in terms of -graded Hochschild cohomology and closed currents in terms of cyclic
cocycles (they are particular multilinear forms on *Gr*(*N*)). An
explicit construction of the vector space of closed currents of
degree *p* on *Gr*(*N*) is given by using Berezin integration.

**Keywords :**

Grassmann algebras, super-algebras, supersymmetries, supermanifolds, Hochschild and cyclic cohomologies, De Rham currents.

October 1993

CPT - 93 / P.2935

ENSLAPP - 440/93

- Introduction
- Differential forms over Grassmann algebras and Grassmannian currents
- Hochschild and cyclic cohomologies for Grassmann algebras
- The action of operators
*S*and*B*in the Grassmannian case - Some combinatorial identities
- References
- About this document ...

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