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

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.