# Kruhlyak S. A.

### Regular orthoscalar representations of the extended Dynkin graph $\widetilde{E}_8$ and ∗-algebra associatedwith it

Kruhlyak S. A., Livins'kyi I. V.

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1044–1062

We obtain a classification of regular orthoscalar representations of the extended Dynkin graph $\widetilde{E}_8$ with special character. Using this classification, we describe triples of self-adjoint operators A, B, and C such that their spectra are contained in the sets $\{0,1,2,3,4,5\}, \{0,2,4\}$, and $\{0,3\}$, respectively, and the equality $A + B + C = 6I$ is true.

### On coxeter functors for some classes of algebras generated by idempotents

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 996–1001

We study properties of annihilation operators of infinite order that act in spaces of test functions. The results obtained are used for establishing the coincidence of spaces of test functions.

### Coxeter Functors for One Class of *-Quivers

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 789-797

For one certain class of *-quivers, we construct Coxeter functors and describe their application to the description of families of orthoprojectors whose sum is a multiple of the identity operator.

### Structure theorems for families of idempotents

Kruhlyak S. A., Samoilenko Yu. S.

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 523–533

For *-algebras generated by idempotents and orthoprojectors, we study the complexity of the problem of description of *-representations to within unitary equivalence. In particular, we prove that the *-algebra generated by two orthogonal idempotents is *-wild as well as the *-algebra generated by three orthoprojectors, two of which are orthogonal.