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# Grassmann-Hopf algebroid categories and Grassmann categories

Grassmann-Hopf Algebroid Categories and Grassmann Categories

###### Definition 0.1.

The categories whose objects are either *Grassmann-Hopf al/gebras*, or in general $G-H$ algebroids,
and whose morphisms are *$G-H$ homomorphisms* are called *Grassmann-Hopf Algebroid Categories*.

Although carrying a similar name, a quite different type of Grassmann categories have been introduced previously:

###### Definition 0.2.

*Grassmann Categories* (as in [1]) are defined *on $k$ letters over nontrivial abelian categories* $\mathbf{\mathcal{A}}$ as *full subcategories* of the categories $F_{{\mathbf{\mathcal{A}}}}(x_{1},...,x_{k})$ consisting of diagrams satisfying the relations: $x_{i}x_{j}+x_{j}x_{i}=0$ and $x_{i}x_{i}=0$ with additional conditions on coadjoints, coproducts and morphisms.

They were shown to be equivalent to the category of right modules over the endomorphism ring of the coadjoint $S(R)$ which is isomorphic to the Grassmann–or exterior–ring over $R$ on $k$ letters $E_{R}(X_{1},...,X_{N})$.

# References

- 1 Barry Mitchell.Theory of Categories., Academic Press: New York and London.(1965), pp. 220-221.
- 2
B. Fauser:
*A treatise on quantum Clifford Algebras*. Konstanz, Habilitationsschrift. (PDF at arXiv.math.QA/0202059).(2002). - 3
B. Fauser: Grade Free product Formulae from Grassmann–Hopf Gebras., Ch. 18 in R. Ablamowicz, Ed.,
*Clifford Algebras: Applications to Mathematics, Physics and Engineering*, Birkhäuser: Boston, Basel and Berlin, (2004). - 4
I.C. Baianu, R. Brown J.F. Glazebrook, and G. Georgescu, Towards Quantum Non-Abelian Algebraic Topology.
*in preparation*, (2008).

## Mathematics Subject Classification

18-00*no label found*18E05

*no label found*18D05

*no label found*15A75

*no label found*

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