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)$ .

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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