Exploring fundamental algebras with involution

Fundamental algebras were essentially introduced by Kemer in his proof of the Specht conjecture in characteristic zero. Here, we extend the concept to algebras equipped with an involution, and we develop the corresponding theory. In our search for examples, we show that not only are simple algebras with involution fundamental, but so is a broader class of algebras, denoted by UT⁎(A1,…,An). These algebras, constructed from upper block triangular matrices, generate varieties of algebras with involution that are minimal with respect to the exponent. Furthermore, we introduce an additional class of fundamental algebras with involution obtained by identifying the simple components of any algebra UT⁎(A1,…,An) in all possible ways.