[Paper Review] Representation embeddings and the second Brauer-Thrall conjecture

Klaus Bongartz|arXiv (Cornell University)|Nov 7, 2016

Algebraic structures and combinatorial models16 references3 citations

TL;DR

This paper establishes a strong representation embedding from the polynomial ring $k[T]$ into any basic, representation-infinite $k$-algebra $A$ of finite dimension $d$, using a bimodule ${}_A M_{k[T]}$ that is free of rank at most $\max(4d, 30)$ over $k[T]$. The key contribution is a numerical version of the second Brauer-Thrall conjecture, proving that such embeddings exist with bounded rank, and classifying minimal embedding classes in both characteristic 0 and positive characteristic.

ABSTRACT

We prove that over an algebraically closed field there is a representation embedding from the category of classical Kronecker-modules without the simple injective into the category of finite-dimensional modules over any representation-infinite finite-dimensional algebra. We also sharpen some known results on representation embeddings, we simplify some proofs and we construct a simultaneous orthogonal embedding for an infinite family of module categories. In the last section the minimal classes of representation-infinite algebras are determined. The result depends on the characteristic.

Motivation & Objective

• To establish a representation embedding from $\mathrm{mod}\,k[T]$ into any basic, representation-infinite $k$-algebra $A$ of finite dimension $d$.
• To prove a numerical form of the second Brauer-Thrall conjecture by bounding the rank of the bimodule inducing the embedding.
• To classify minimal embedding classes of representation-infinite algebras in both characteristic 0 and positive characteristic.
• To clarify the structure of embedding classes, especially in relation to wild and tame algebras.
• To provide new interpretations and proofs of classical results in representation theory using the framework of representation embeddings.

Proposed method

• Constructs a bimodule ${}_A M_{k[T]}$ that is free of rank at most $\max(4d, 30)$ over $k[T]$, where $A$ is a basic, representation-infinite $k$-algebra of dimension $d$.
• Uses tensoring with this bimodule as the representation embedding $F: \mathrm{mod}\,k[T] \to \mathrm{mod}\,A$
• Applies deep theorems from representation theory, including results on preprojective components and tilting theory, to analyze the structure of the embedding.
• Employs elementary set theory and matrix-free constructions to build embeddings from $\mathrm{mod}\,A_i$ to $\mathrm{mod}\,kQ$ for a wild quiver $Q$ with pairwise orthogonal hom-spaces.
• Utilizes the Eilenberg-Watts theorem to represent exact, faithful functors as tensor functors with bimodules.
• Analyzes Morita equivalence classes and embedding classes via properties of preprojective and preinjective components in quiver representations.

Experimental results

Research questions

• RQ1 Can every basic, representation-infinite $k$-algebra $A$ of finite dimension $d$ be embedded into $\mathrm{mod}\,k[T]$ via a bimodule that is free of bounded rank over $k[T]$?
• RQ2 What is the minimal possible rank of such a bimodule, and how does it depend on $d$?
• RQ3 How do embedding classes of representation-infinite algebras decompose in characteristic 0 versus positive characteristic?
• RQ4 Are there only finitely many Morita-equivalence classes in the embedding class of $kK_2$ in positive characteristic?
• RQ5 What is the structure of the embedding class of $k[X,Y]/(X,Y)^2$ in characteristic 0, and how does it compare to that of $kK_2$?

Key findings

• For any basic, representation-infinite $k$-algebra $A$ of dimension $d$, there exists a bimodule ${}_A M_{k[T]}$ free of rank at most $\max(4d, 30)$ over $k[T]$ such that tensoring with $M$ gives a representation embedding $\mathrm{mod}\,k[T] \to \mathrm{mod}\,A$.
• In characteristic 0, the minimal embedding classes are those of $kK_2$ and $k[X,Y]/(X,Y)^2$, each containing exactly one Morita-equivalence class.
• In positive characteristic $p > 0$, the only minimal embedding class is that of $kK_2$, but it contains countably many Morita-equivalence classes.
• Any representation embedding $F: \mathrm{mod}\,A \to \mathrm{mod}\,kK_2$ with $A$ representation-infinite implies $A$ is Morita-equivalent to $kK_2 \times B$ with $B$ representation-finite.
• The embedding class of $kK_2$ in positive characteristic contains only countably many Morita-equivalence classes, due to the existence of multiplicative bases for representation-finite algebras.
• Two algebras of the form $k\langle X,Y\rangle/(X^2,Y^2,XY-\lambda YX)$ are in the same embedding class if and only if they are isomorphic, implying infinitely many distinct embedding classes for tame algebras.