Operator representations of \(\mathcal{U}_ q(sl_ 2(\mathbb{R}))\)
Let \(q\) be a complex number, different from 0, \(-1\), and 1. The quantized universal enveloping algebra \(U_q (sl(2))\) can be described as the algebra over \(\mathbb{C}\) with unit generated by an invertible element \(K\) and two elements \(X_+\), \(X_-\), satisfying the commutation rules \[ KK_+ =qX_+ K, \quad KX_- =q^{-1} X_-K, \quad X_+ X_-- X_-X_+ = (q- q^{-1})^{-1} (K^2- K^{-2}). \] This algebra was first introduced by \textit{P. Kulish} and \textit{N. Reshetikhin} [J. Sov. Math. 23, 2435-2441 (1983); translation from Zap. Nauchn. LOMI 101, 101-110 (1981)]. It carries a natural comultiplication making it into a Hopf algebra. Then, it is the simplest example of the quantum groups introduced by \textit{V. Drinfel'd} [Proc. Int. Congr. Math., Berkeley 1986, Vol 1, 798-820 (1987; Zbl 0476.16008)] and \textit{M. Jimbo} [Lett. Math. Phys. 10, 63-69 (1985; Zbl 0602.17005)]. There are several natural \(*\)-structures on this algebra (for associated special values of \(q)\) making this Hopf algebra into a Hopf \(*\)-algebra. The best known one is given by \(K^* =K\) and \(X_+^* = X_-\). Here one must have \(q\in \mathbb{R}\). This gives the quantization of \(su(2)\). For \(|q |= 1\), we can take \(K^* =K\) and \(X_+^*= -X_+\), \(X^*_- = -X_-\). This is considered as the quantization of \(sl(2,\mathbb{R})\). The first one gives a compact quantum group while the second one yields a non-compact quantum group. In the first case, it is not so hard to construct operator representations on Hilbert spaces (respecting the \(*\)-structure) because there are finite-dimensional representations (hence with bounded operators). In the second case, it is much more subtle to get representations because the operators are necessarily unbounded (in the nontrivial case) and in particular, one has to say precisely what is meant by an operator representation of this algebra. In this paper, the author defines and studies certain types of operator representations of the \(*\)-algebra \(U_q (sl (2, \mathbb{R}))\) with \(|q |= 1\). They are called integrable representations following the terminology which is commonly used in representation theory of Lie algebras etc. The basic observation is the following. Let \(A= iX_+\) and \(B= iX_-\), then \(A\) and \(B\) are symmetric elements and they satisfy a similar commutation rule w.r.t. \(K\), namely \[ KA = qAK \quad \text{and} \quad KB = q^{-1} BK. \] So, we get two copies of the \(*\)-algebra generated by two self-adjoint elements \(a\) and \(b\) satisfying \(ab = qba\). This algebra is generally considered as the quantum plane \(\mathbb{R}^2_q\). The integral operator representations of this algebra have been studied in [Commun. Math. Phys. 159, 217-237 (1994; Zbl 0822.46064) by the same author. Two classes are obtained and the essential building block for the two cases is given by the pairs of operators \(e^{\alpha x}\) and \(e^{\beta p}\) where \(x\) and \(p\) are the standard self-adjoint operators on \(L^2 (\mathbb{R})\) satisfying the commutation rule \([x, p] = i\) and where \(\alpha\) and \(\beta\) are real numbers such that \(e^{i \alpha \beta} = q\). The representations of the \(*\)-algebra \(U_q (sl(2, \mathbb{R}))\) considered in this paper are such that the pairs \((K,A)\) and \((K,B)\) are of the above type and such that moreover the operators \(K\), \(A\) and \(B\) strongly commute with the self-adjoint operator \(C\), representing the Casimir element \[ BA- (q- q^{-1})^{-2} (qK^2 + q^{-1} K^{-2}). \] Finally, two other classes of operator representations are constructed where the elements \(A\) and \(B\) are represented by symmetric, but non-self-adjoint operators. They give an interesting link between the representations of the algebra \(U_q (sl (2, \mathbb{R}))\) and complex function theory on a strip in the complex plane.
scientific article; zbMATH DE number 888968