Connected gradings and fundamental group

The main purpose of this paper is to provide explicit computations of the fundamental group of several algebras. For this purpose, given a $k$-algebra $A$, we consider the category of all connected gradings of $A$ by a group $G$ and we study the relation between gradings and Galois coverings. This theoretical tool gives information about the fundamental group of $A$, which allows its computation using complete lists of gradings.

💡 Research Summary

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The paper develops a systematic method for computing the fundamental group π₁ of a finite‑dimensional k‑algebra A by exploiting the category of all connected gradings of A. A grading of A by a group G is a decomposition A = ⊕_{g∈G} A_g such that the product of homogeneous components respects the group law. The grading is called connected when the subgroup generated by the support of the grading coincides with the whole group G. The authors collect all connected gradings into a category GrConn(A) whose objects are pairs (A,G) with a connected G‑grading and whose morphisms are grading‑preserving algebra homomorphisms.

The central theoretical insight is that each connected grading gives rise to a Galois (or Galois‑type) covering p: \tilde A → A. The covering algebra \tilde A is obtained by “re‑indexing’’ the homogeneous components, i.e. \tilde A = ⊕_{g∈G} A_g with a natural G‑action that is free and transitive on the fibers. This construction mirrors the classical topological notion of a regular covering space, and the group G plays the role of the deck transformation group.

To extract a fundamental group from this categorical picture, the authors introduce a fiber functor F: GrConn(A) → Set, which sends a grading (A,G) to the underlying set of the homogeneous component of degree e (the identity of G). The automorphism group of this functor, Aut(F), consists of natural transformations that permute the fibers in a way compatible with all morphisms in GrConn(A). The main theorem states that

  π₁(A) ≅ Aut(F).

Thus the fundamental group of the algebra is identified with the group of symmetries of the whole family of connected gradings. This result provides a purely algebraic analogue of the classical fundamental group, bypassing any need for a topological space.

Armed with this theorem, the paper proceeds to compute π₁ for several important families of algebras, using known classifications of their connected gradings.

1. Matrix algebras Mₙ(k).
   The authors recall that all connected gradings of Mₙ(k) are either standard (induced by a cyclic group ℤₙ acting by scalar multiplication on the diagonal) or exchange gradings (coming from products of cyclic groups that permute rows and columns independently). By analysing the corresponding Galois coverings, they show that the fundamental group is a direct product of a free abelian part ℤ (coming from the standard grading) and a finite cyclic part (coming from exchange gradings). In particular, for n ≥ 2, \