Description of the structure of DG--algebras via characters on a groupoid

TL;DR

This work addresses constructing DG-algebra structures on graded group algebras by translating graded derivations into locally finite characters on a groupoid of conjugacy action. It builds a groupoid $\Gamma$, proves $\operatorname{Der}(\mathbb{C}[G]) \cong \chi(\Gamma)$, and classifies inner, central, and quasi-inner derivations through these characters. It then gives necessary and sufficient conditions for a character to define a DG-algebra structure and an explicit isomorphism criterion for DG-algebras, with concrete examples on groups such as the Heisenberg group. The approach unifies derivations and DG-structures in graded group algebras and offers a practical method to construct DG-algebras using combinatorial group theory tools.

Abstract

We construct a description of graded derivations in group algebras. Using this result for arbitrary graduation of the group algebra, we describe all possible structures of DG algebras. The corresponding examples are given. The description is given in terms of characters on a groupoid analogous to the groupoid of inner action.

Theorems & Definitions (38)

• definition 1
• definition 2
• proposition 1
• proof
• proposition 2
• definition 3
• definition 4
• definition 5
• proposition 3
• proof