Representations of the \(q\)-deformed algebra \(\text{so}_q(2,1)\) (Q2711725)

The authors give a classification of irreducible weight representations of the \(q\)-deformed algebra \(U_q(\text{so}_{2,1})\), a real form of the nonstandard deformation \(U_q(\text{so}_3)\) of the Lie algebra \(\text{so}(3,\mathbb{C})\). The algebra \(U_q(\text{so}_3)\) is generated by the elements \(I_1,I_2,I_3\) satisfying the relations \([I_1,I_2]_q\equiv q^{1/2}I_1I_2-q^{-1/2}I_2I_1=I_3\), \([I_2, I_3]_q=I_1\), \([I_3,I_1]_q=I_2\). The real form \(U_q(\text{so}_{2,1})\) is determined for a real \(q\) by the involution \(I_1^*=-I_1\), \(I_2^*=I_2\). Weight representations of \(U_q(\text{so}_{2,1})\) are those representations \(T\) for which \(T(I_1)\) can be diagonalized and has discrete spectrum. The behaviour of representations as \(q\to 1\) is studied.NEWLINENEWLINEFor the entire collection see [Zbl 0937.00046].