[Paper Review] Complete special biserial algebras are $g$-tame

Toshitaka Aoki, Toshiya Yurikusa|arXiv (Cornell University)|Mar 22, 2020

Algebraic structures and combinatorial models23 references4 citations

TL;DR

This paper proves that complete special biserial algebras are $g$-tame by leveraging their surface model and asymptotic behavior under Dehn twists, showing that the closure of the $g$-vector fan spans the entire ambient vector space. The result follows from the fact that complete gentle algebras are $g$-tame and $g$-tameness is preserved under factor algebras.

ABSTRACT

The $g$-vectors of two-term presilting complexes are important invariant. We study a fan consisting of all $g$-vector cones for a complete gentle algebra. We show that any complete gentle algebra is $g$-tame, by definition, the closure of a geometric realization of its fan is the entire ambient vector space. Our main ingredients are their surface model and their asymptotic behavior under Dehn twists. On the other hand, it is known that any complete special biserial algebra is a factor algebra of a complete gentle algebra and the $g$-tameness is preserved under taking factor algebras. As a consequence, we get the $g$-tameness of complete special biserial algebras.

Motivation & Objective

• To establish $g$-tameness for complete special biserial algebras, a key property in representation theory.
• To extend the known $g$-tameness of complete gentle algebras to a broader class of algebras via factor algebra construction.
• To analyze the geometric structure of $g$-vector cones using surface models and topological dynamics.
• To understand the asymptotic behavior of $g$-vectors under Dehn twists in the context of surface algebras.
• To demonstrate that the closure of the $g$-vector fan fills the entire ambient vector space, confirming $g$-tameness.

Proposed method

• Utilize the surface model of complete gentle algebras to geometrically represent $g$-vectors as cones in a vector space.
• Analyze the action of Dehn twists on the surface model to study the asymptotic behavior of $g$-vectors.
• Construct the fan of $g$-vector cones from two-term presilting complexes over complete gentle algebras.
• Prove that the closure of this fan spans the entire ambient vector space, establishing $g$-tameness for complete gentle algebras.
• Apply the preservation of $g$-tameness under factor algebras to extend the result to complete special biserial algebras.
• Leverage the fact that every complete special biserial algebra is a quotient of a complete gentle algebra to transfer the $g$-tameness property.

Experimental results

Research questions

• RQ1 Are complete special biserial algebras $g$-tame, meaning does the closure of their $g$-vector fan span the ambient space?
• RQ2 How does the surface model of a complete gentle algebra facilitate the analysis of $g$-vector cones and their asymptotic behavior?
• RQ3 What role do Dehn twists play in the geometric realization of $g$-vectors and the spanning property of the fan?
• RQ4 Does $g$-tameness persist when passing from a complete gentle algebra to its factor algebra?
• RQ5 Can the $g$-vector fan of a complete special biserial algebra be shown to fill the entire ambient vector space through structural and topological arguments?

Key findings

• Complete gentle algebras are $g$-tame, as the closure of their $g$-vector fan spans the entire ambient vector space.
• The asymptotic behavior of $g$-vectors under Dehn twists on the surface model ensures dense distribution of cone directions.
• The fan of $g$-vector cones for complete gentle algebras is geometrically realized in such a way that its closure fills the ambient space.
• Every complete special biserial algebra arises as a factor algebra of a complete gentle algebra.
• The property of $g$-tameness is preserved under taking factor algebras, allowing the extension of $g$-tameness to the broader class.
• As a consequence, all complete special biserial algebras are $g$-tame, completing the classification for this class of algebras.