algebras of sectorial, radial and boundary oscillators are associative algebras, which can be (minimally) extended to some (infinite-dimensional) Lie algebras. The construction of such extensions requires the introduction of additional generators satisfying the new commutation relations induced by commutation relations of considered associative algebras. These constructions are the main content of Sec.3—Sec.5.

Finally, in section 6, is determined the Lie algebra of two-dimensional ChK-oscillator. Note that it is more natural to understood as oscillator algebra not the associative algebra but its minimal extension to the Lie algebra. We suppose that this is correct because the considered algebra should be a generalization of the Heisenberg-Lie algebra for the quantum-mechanical oscillator. Some properties of this infinite-dimensional Lie algebra are investigated in Sec.7.

Preliminaries
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For the reader convenience, we present briefly some information (necessary for further discussions ) from the works [@N46; @N47].

The Chebyshev — Koornwinder polynomials (ChK-polynomials)
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The Chebyshev — Koornwinder polynomials (ChK-polynomials) [@N28] can be defined by recurrent relations $$\begin{aligned}
\label{13-1.1}\index{1}
z\,U_{k,\,l}(z,\,\z)=U_{k+1,\,l}(z,\,\z)+U_{k,\,l-1}(z,\,\z)+U_{k-1,\,l+1}(z,\,\z),\nn\\
\z\,U_{k,\,l}(z,\,\z)=U_{k,\,l+1}(z,\,\z)+U_{k-1,\,l}(z,\,\z)+U_{k+1,\,l-1}(z,\,\z),\end{aligned}$$ subject to the following conditions $$U_{0,\,-1}\!=\!U_{-1,\,0}\!=\!0,\,U_{0,\,0}\!=\!1,\quad
\overline{U_{k,\,l}}(z,\,\z)\!=\!U_{l,\,k}(z,\,\z)\!=\!U_{k,\,l}(\z,\,z).$$ It is known [@N28] that ChK-polynomials form an orthonormal basis in the Hilbert space $\H=\L^2(S;\mu(\d x\d y)),$ where $S$ — the region within the Steiner hypocycloid (see Fig.1), 