Categories and weak equivalences of graded algebras

When one studies the structure (e.g. graded ideals, graded subspaces, radicals, ...) or graded polynomial identities of graded algebras, the grading group itself does not play an important role, but can be replaced by any other group that realizes the same grading. Here we come to the notion of weak equivalence of gradings: two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component. Each group grading on an algebra can be weakly equivalent to G-gradings for many different groups G, however it turns out that there is one distinguished group among them called the universal group of the grading. In this paper we study categories and functors related to the notion of weak equivalence of gradings. In particular, we introduce an oplax 2-functor that assigns to each grading its support and show that the universal grading group functor has neither left nor right adjoint.Comment: 17 pages; a preliminary version of this article was previously a part of arXiv:1704.07170. (To appear in Algebra Colloquium.

Quotient gradings and the intrinsic fundamental group

Quotient grading classes are essential participants in the computation of the intrinsic fundamental group $\pi_1(A)$ of an algebra $A$. In order to study quotient gradings of a finite-dimensional semisimple complex algebra $A$ it is sufficient to understand the quotient gradings of twisted gradings. We establish the graded structure of such quotients using Mackey's obstruction class. Then, for matrix algebras $A=M_n(\mathbb{C})$ we tie up the concepts of braces, group-theoretic Lagrangians and elementary crossed products. We also manage to compute the intrinsic fundamental group of the diagonal algebras $A=\mathbb{C} ^4$ and $A=\mathbb{C} ^5$.Comment: 33 page

Units of twisted group rings and their correlations to classical group rings

Given a central extension $\Gamma$ of some normal subgroup $N$ by a group $G$, we study the group ring $R \Gamma$ over some domain $R$. We obtain a direct sum decomposition in terms of various twisted group rings of $G$ and concrete information on the kernel and cokernel of the projections. This allows to kick-start the investigation of the unit group $\mathcal{U}( R\Gamma)$ via the unit group of twisted group rings. Among others we construct a new generic family of units therein. As an application hereof we are able to obtain a complete description of the unit group of the integral group ring of $G \times C_2^n$ from data of $G$.Comment: Preliminary version, 26 page