A majorization relation for a certain class of *-quivers with an orthogonality condition

In [Kruglyak S. A., Samoĭlenko Yu. S.: On structure theorems for a family of idempotents. Ukrainskii Matematicheskii Zhurnal 50 (4), (1998) (Russian), Kruglyak S. A., Samoĭlenko Yu. S.: On the complexity of description of representations of \(*\)-algebras generated by idempotents. [in:] Proc. of the American Mathematical Society, vol. 128, 1655–1664, AMS, 2000, Kruglyak S. A.: A majorization relation for \(*\)-categories and \(*\)-wild categories. [in:] Proc. of the Fifth Intern. Conference "Symmetry in Nonlinear Math. Physics", 2004], \(*\)-algebras and \(*\)-categories over the field \(\mathbb{C}\) of complex numbers were quasi-ordered with respect to the complexity of the structure of their \(*\)-representations with a majorization relation \(\succ\). A notion of \(*\)-wildness was also introduced there for an algebra (a category) if the algebra majorizes the \(*\)-algebra \(C^{*}(\mathcal{F}_2)\). In this paper, we discuss some methods for proving that an algebra is \(*\)-wild and obtain criteria for certain "standard" \(*\)-categories (ensembles with an orthogonality condition) to be \(*\)-wild.