0.1 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 𝒜 as full subcategories of the categories F𝒜(x1,...,xk) consisting of
diagrams satisfying the relations: xixj +xjxi = 0 and xixi = 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 ER(X1,...,XN).
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).
