Representations of the \(q\)-deformed Lie algebra of the group of motions of the Euclidean plane
Let \(\{M_q\}\) be a family of complex unital associative algebras with generators \(z_1\), \(z_2\), \(z_3\) and relations \[ q^{1/2}z_1z_2-q^{-1/2}z_2z_1 =z_3,\quad q^{1/2} z_2z_3-q^{-1/2} z_3z_2=z_1, \quad q^{1/2}z_3z_1-q^{-1/2} z_1 z_3=0,\;q\in\mathbb{C}. \] In this paper a Hilbert space representation of the real forms of the algebra \(M_q\) is considered which are defined an involution \(*:M_q\to M_q\). The authors define and classify ``well-behaved'' representations for real forms of the algebra \(M_q\) defined by linear and some quadratic involutions. Using the results of \textit{V. Ostrovskyi} and \textit{Yu. Samoilenko} [Rep. Math. Phys. 28, 91-103 (1989; Zbl 0749.47020)] they present such a definition for the \(*\)-algebras \(M_q^{2,1}\), \(M_q^{2,-1}\) and give a description of all irreducible ``well-behaved'' representations up to unitary equivalence. It is shown that the problem of classification of all irreducible representations defined on some dense invariant domain is very difficult even in very simple cases. It is proved that in order to classify all such representations it is necessary also to classify all finite families of (selfadjoint) Hilbert space operators up to unitary equivalence. In this paper it is proved that such a classification can be achieved for some subclasses of representations of \(M^1_q\) and \(M_q^{-1}\).
scientific article; zbMATH DE number 1246378
scientific article