2019
Том 71
№ 11

# Livins'kyi I. V.

Articles: 3
Article (Ukrainian)

### Representations of Algebras Defined by a Multiplicative Relation and Corresponding to the Extended Dynkin Graphs $\tilde{D}_4, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1654-1675

We describe, up to unitary equivalence, all $k$-tuples $(A_1, A_2,..., A_k)$ of unitary operators such that $A^{n_i}_i = I$ for $i = \overline{1, k}$ and $A_1 A_2 ... A_k = \lambda I$, where the parameters $(n_1,... ,n_k)$ correspond to one of the extended Dynkin diagrams $\tilde{D}_4, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$, and $\lambda \in \mathbb{C}$ is a fixed root of unity.

Article (Ukrainian)

### Regular orthoscalar representations of extended dynkin graphs $\widetilde{E}_6$ and $\widetilde{E}_7$ and *-algebras associatedwith them

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1459–1472

We obtain a classification of indecomposable orthoscalar representations of the extended Dynkin graphs $\widetilde{E}_6$ and $\widetilde{E}_7$ with a special character and of the *-algebras associated with them, up to the unitary equivalence.

Article (Ukrainian)

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

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.