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

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 $E_R(X_1,..., X_N)$ .

Bibliography

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).

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