On semisimple varietal Terwilliger algebras whose non-primary ideals are 1-dimensional

As an abstraction of the Terwilliger algebra of a commutative association scheme, the notion of a varietal Terwilliger algebra was introduced in [11]. In this paper we first prove that there exists a varietal Terwilliger algebra defined by any given standard C-algebra, and a generalized Terwilliger algebra [6] is also a varietal Terwilliger algebra. Then we present a necessary and sufficient condition under which a quotient algebra of a varietal Terwilliger algebra is also a varietal Terwilliger algebra. We will also give a characterization of a semisimple varietal Terwilliger algebra whose non-primary ideals are all 1-dimensional, and prove that a semisimple regular varietal Terwilliger algebra with 1-dimensional non-primary ideals is characterized by the structure of its base C-subalgebra. As direct consequences, we obtain some results in [18] for finite association schemes and a conceptual explanation of the results in [9].