Representations of the q-deformed algebra Uq(iso2)U_q({\rm iso}_2)

An algebra homomorphism $\psi$ from the q-deformed algebra $U_q({\rm iso}_2)$ with generating elements $I$, $T_1$, $T_2$ and defining relations $[I,T_2]_q=T_1$, $[T_1,I]_q=T_2$, $[T_2,T_1]_q=0$ (where $[A,B]_q=q^{1/2}AB-q^{-1/2}BA$) to the extension ${\hat U}_q({\rm m}_2)$ of the Hopf algebra $U_q({\rm m}_2)$ is constructed. The algebra $U_q({\rm iso}_2)$ at $q=1$ leads to the Lie algebra ${\rm iso}_2 \sim {\rm m}_2$ of the group ISO(2) of motions of the Euclidean plane. The Hopf algebra $U_q({\rm m}_2)$ is treated as a Hopf $q$-deformation of the universal enveloping algebra of ${\rm iso}_2$ and is well-known in the literature. Not all irreducible representations of $U_q({\rm m}_2)$ can be extended to representations of the extension ${\hat U}_q({\rm m}_2)$. Composing the homomorphism $\psi$ with irreducible representations of ${\hat U}_q({\rm m}_2)$ we obtain representations of $U_q({\rm iso}_2)$. Not all of these representations of $U_q({\rm iso}_2)$ are irreducible. The reducible representations of $U_q({\rm iso}_2)$ are decomposed into irreducible components. In this way we obtain all irreducible representations of $U_q({\rm iso}_2)$ when $q$ is not a root of unity. A part of these representations turns into irreducible representations of the Lie algebra iso$_2$ when $q\to 1$. Representations of the other part have no classical analogue.Comment: 12 pages, LaTe

Професор Г.Л. Кулініч (09.12.1938 – 10.02.2022) – видатний вчений і педагог

Pages of the article in the issue: 11 - 21 Language of the article: UkrainianСторінки у випуску: 11 - 21 Мова статті: українськ

Measurement of the K+→μ+νμγK^+\rightarrow{\mu^+}{\nu_{\mu}}{\gamma} decay form factors in the OKA experiment

A precise measurement of the vector and axial-vector form factors difference $F_V-F_A$ in the $K^+\rightarrow{\mu^+}{\nu_{\mu}}{\gamma}$ decay is presented. About 95K events of $K^+\rightarrow{\mu^+}{\nu_{\mu}}{\gamma}$ are selected in the OKA experiment. The result is $F_V-F_A=0.134\pm0.021(stat)\pm0.027(syst)$. Both errors are smaller than in the previous $F_V-F_A$ measurements.Comment: 9 pages, 8 figure

Searches for the light invisible axion-like particle in K+→π+π0aK^{+}\to\pi^{+}\pi^{0}a decay

A high statistics data sample of the $K^{+}$ decays is recorded by the OKA collaboration. A missing mass analysis is performed to search for a light invisible pseudoscalar axion-like particle (ALP) $a$ in the decay $K^{+} \to \pi^{+} \pi^{0} a$. No signal is observed, the upper limits for the branching ratio of the decay are calculated. The $90\%$ confidence level upper limit is changing from $2.5\cdot10^{-6}$ to $2\cdot10^{-7}$ for the ALP mass from 0 to 200 MeV/$c^{2}$, except for the region of $\pi^{0}$ mass, where the upper limit is $4.4\cdot10^{-6}$.Comment: 6 pages, 6 figure

Study of the decay K+→π+π−π+γK^{+}\to\pi^{+}\pi^{-}\pi^{+}\gamma in the OKA experiment

A high statistics data sample of the decays of $K^+$ mesons to three charged particles was accumulated by the OKA experiment in 2012 and 2013. This allowed to select a clean sample of about 450 events with $K^{+}\to\pi^{+}\pi^{-}\pi^{+}\gamma$ decays with the energy of the photon in the kaon rest frame greater than 30 MeV. The measured branching fraction of the $K^{+}\to\pi^{+}\pi^{-}\pi^{+}\gamma$, with $E_{\gamma}^{*}$ > 30 MeV is $(0.71 \pm 0.05) \times 10^{-5}$. The measured energy spectrum of the decay photon is compared with the prediction of the chiral perturbation theory to O$(p^{4})$. A search for an up-down asymmetry of the photon with respect to the hadronic system decay plane is also performed.Comment: 9 pages, 7 figures, 1 tabl