[Paper Review] Structure Theorems for Basic Algebras

Carl Fredrik Berg|arXiv (Cornell University)|Feb 5, 2011

Algebraic structures and combinatorial models12 references3 citations

TL;DR

This paper generalizes Gabriel's structure theorem for finite-dimensional basic algebras over algebraically closed fields to arbitrary fields, particularly non-perfect fields. It introduces species with relations to describe module categories and shows that when the radical filtration sequence fails to split, the algebra becomes a quotient of a tensor algebra by a non-admissible ideal—leading to hereditary algebras that are not tensor algebras, with non-zero relations arising from cyclic subgraphs in the underlying species graph.

ABSTRACT

A basic finite dimensional algebra over an algebraically closed field $k$ is isomorphic to a quotient of a tensor algebra by an admissible ideal. The category of left modules over the algebra is isomorphic to the category of representations of a finite quiver with relations. In this article we will remove the assumption that $k$ is algebraically closed to look at both perfect and non-perfect fields. We will introduce the notion of species with relations to describe the category of left modules over such algebras. If the field is not perfect, then the algebra is isomorphic to a quotient of a tensor algebra by an ideal that is no longer admissible in general. This gives hereditary algebras isomorphic to a quotient of a tensor algebra by a non-zero ideal. We will show that these non-zero ideals correspond to cyclic subgraphs of the graph associated to the species of the algebra. This will lead to the ideal being zero in the case when the underlying graph of the algebra is a tree.

Motivation & Objective

• To extend Gabriel’s structure theorem for basic finite-dimensional algebras from algebraically closed fields to arbitrary fields, especially non-perfect fields.
• To investigate the failure of the radical filtration sequence to split in non-perfect fields and its consequences on algebra structure.
• To develop a framework using species with relations to describe the category of left modules over basic finite-dimensional split algebras over non-perfect fields.
• To characterize when a hereditary basic finite-dimensional split algebra over a non-perfect field is isomorphic to a tensor algebra, identifying the role of cyclic subgraphs in the underlying graph.
• To provide a counterexample showing that hereditary algebras over non-perfect fields need not be isomorphic to tensor algebras, even when split.

Proposed method

• Introduce the concept of a species with relations, generalizing quiver representations to non-perfect fields using division rings and bimodules.
• Construct a tensor algebra from a species and define a surjective algebra homomorphism onto the original algebra, even when the kernel is not admissible.
• Use the failure of the radical filtration sequence to split as a bimodule sequence to identify non-admissible ideals in the tensor algebra quotient.
• Characterize the kernel of the tensor algebra map in terms of cyclic subgraphs in the underlying graph of the species, showing that non-zero relations arise precisely when cycles are present.
• Apply the theory to hereditary algebras, proving that such algebras are isomorphic to a tensor algebra quotient only when the underlying graph is a tree.
• Use an explicit example over a non-perfect field $\mathbb{F}_2(t^2)$ to demonstrate a hereditary algebra that is split but not $\mathfrak{r}$-split, hence not a tensor algebra.

Experimental results

Research questions

• RQ1 How does the structure theorem for basic finite-dimensional algebras over algebraically closed fields extend to non-perfect fields?
• RQ2 What conditions on the radical filtration sequence determine whether a basic algebra over a non-perfect field is isomorphic to a tensor algebra?
• RQ3 In what way do cyclic subgraphs in the underlying graph of a species give rise to non-trivial relations in the algebra structure?
• RQ4 Can a hereditary basic finite-dimensional split algebra over a non-perfect field fail to be isomorphic to a tensor algebra?
• RQ5 What role does the splitting of the radical filtration sequence play in determining the algebraic structure of basic algebras over arbitrary fields?

Key findings

• When the field $k$ is not algebraically closed, a basic finite-dimensional split algebra $\Lambda$ is isomorphic to a quotient of a tensor algebra over a species, but the kernel is not necessarily admissible.
• The algebra $\Lambda$ is isomorphic to a tensor algebra quotient if and only if the underlying graph of the species is a tree, in which case all canonical quivers are trivial.
• For hereditary algebras over non-perfect fields, the presence of cycles in the underlying graph of the species leads to non-zero relations in the algebra, even when the algebra is split.
• The example over $k = \mathbb{F}_2(t^2)$ constructs a hereditary algebra $\Lambda$ that is split but not $\mathfrak{r}$-split, hence not isomorphic to a tensor algebra.
• The sequence $0 \to \mathfrak{r}^2 \to \mathfrak{r} \to \mathfrak{r}/\mathfrak{r}^2 \to 0$ fails to split as a bimodule sequence in the example, confirming the kernel is not admissible.
• When the underlying graph of the species is a tree, the algebra $\Lambda$ is isomorphic to both $T(\mathcal{S}_m)$ and $T(\mathcal{S}_\Lambda)$, confirming the structure theorem holds in this case.