Quotient gradings and the intrinsic fundamental group

TL;DR

This paper explores quotient gradings and their role in computing the intrinsic fundamental group of algebras, establishing structural results and explicit calculations for certain algebra classes.

Contribution

It provides a detailed analysis of quotient gradings of twisted gradings using Mackey's obstruction, and computes the fundamental group for specific diagonal algebras.

Findings

Established graded structure of quotient gradings via Mackey's class

Connected braces, Lagrangians, and crossed products in matrix algebras

Computed the fundamental group for 4 and 5 algebras

Abstract

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 $A=\mathbb{C} ^5$.