On Terwilliger $\mathbb{F}$-algebras of quasi-thin association schemes

TL;DR

This paper investigates the algebraic properties of Terwilliger algebras over fields for quasi-thin association schemes, including their dimensions, semisimplicity, radicals, and structure, advancing understanding in algebraic combinatorics.

Contribution

It determines key algebraic properties of Terwilliger $ extbf{F}$-algebras for quasi-thin schemes, extending prior definitions to fields and providing new structural insights.

Findings

Calculated the $ extbf{F}$-dimensions of the algebras

Established conditions for semisimplicity and radicals

Described the algebraic structures of the Terwilliger $ extbf{F}$-algebras

Abstract

In [3], Hanaki defined the Terwilliger algebras of association schemes over a commutative unital ring. In this paper, we call the Terwilliger algebras of association schemes over a field $\mathbb{F}$ the Terwilliger $\mathbb{F}$-algebras of association schemes and study the Terwilliger $\mathbb{F}$-algebras of quasi-thin association schemes. As main results, we determine the $\mathbb{F}$-dimensions, the semisimplicity, the Jacobson radicals, and the algebraic structures of the Terwilliger $\mathbb{F}$-algebras of quasi-thin association schemes. We also get some results with independent interests.