# Chapovsky Yu.

### Construction of continuous cocycles for the bicrossed product of locally compact groups

Chapovsky Yu., Podkolzin G. B.

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 243-260

For locally compact groups $K, M$, and $N$ such that $M$ and $N$ are subgroups of $K, K = M ∙ N$ and $M ∩ N = \{e\}$, where $e$ is the identity of the group $K$, we give a complete description and propose a method for the construction of pairs of continuous cocycles used in the structure of bicrossed product with cocycles in terms of continuous 2-cocycles on the groups $M, N$, and $$K and 3-cocycles on the group $K$.

### Finding cocycles in the bicrossed product construction for Lie groups

Chapovsky Yu., Kalyuzhnyi A. A., Podkolzin G. B.

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1510–1522

We find an explicit formula for finding pairs of cocycles for the construction of examples of locally compact quantum groups by using the bicrossed product of Lie groups.

### Conditional Expectations on Compact Quantum Groups and Quantum Double Cosets

Chapovsky Yu., Kalyuzhnyi A. A., Podkolzin G. B.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 644–653

We prove that a conditional expectation on a compact quantum group that satisfies certain conditions can be decomposed into a composition of two conditional expectations one of which is associated with quantum double cosets and the other preserves the counit.

### Factorization of Conditional Expectations on Kac Algebras and Quantum Double Coset Hypergroups

Chapovsky Yu., Kalyuzhnyi A. A.

Ukr. Mat. Zh. - 2003. - 55, № 12. - pp. 1669-1677

We prove that a conditional expectation on a Kac algebra, under certain conditions, decomposes into a composition of two conditional expectations of a special type and gives rise to a compact quantum hypergroup connected to a quantum Gelfand pair.

### Gelfand pair associated with a hoph algebra and a coideal

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1055–1066

We consider a pair of a compact quantum group and a coideal in its dual Hopf *-algebra and introduce the notions of Gelfand pair and strict Gelfand pair. For a strict Gelfand pair, we construct two hyper-complex systems dual to each other. As an example, we consider the quantum analog of the pair (U*(n), SO(n)*).