# PlanetPhysics/Grassmann Hopf Algebroid Categories and Grassmann Categories

\newcommand{\sqdiagram}[9]{$\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}$ }

### Grassmann-Hopf Algebroid Categories and Grassmann Categories

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

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

Grassmann Categories (as in [1]) are defined on ${\displaystyle k}$ letters over nontrivial abelian categories $\displaystyle \mathbf{\A}$ as full subcategories of the categories $\displaystyle F_{\mathbf{\A}}(x_1,...,x_k)$ consisting of diagrams satisfying the relations: ${\displaystyle x_{i}x_{j}+x_{j}x_{i}=0}$ and ${\displaystyle x_{i}x_{i}=0}$ 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 ${\displaystyle S(R)}$ which is isomorphic to the Grassmann--or exterior--ring over ${\displaystyle R}$ on ${\displaystyle k}$ letters ${\displaystyle E_{R}(X_{1},...,X_{N})}$.

## 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 , Birkh\"{a}user: 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).