On the theory of quantum groups
Some time ago \textit{S. L. Woronowicz} [Publ. Res. Inst. Math. Sci. 23, 117-181 (1987; Zbl 0676.46050)]; Commun. Math. Phys. 111, 613-665 (1987; Zbl 0627.58034)] introduced a one-parameter family of compact matrix quantum groups \(SU_{\mu}(N+1)\), \(\mu\in (0,1]\). Later \textit{M. Rosso} [C. R. Acad. Sci., Paris, Sér. I 304, 323-326 (1987; Zbl 0617.16005)] showed that \(SU_{\mu}(N+1)\) is related by duality to the q-deformation \(U_ q(su(N+1))\) of the universal enveloping algebra \(U(su(N+1))\) introduced independently by \textit{V. G. Drinfel'd} [Dokl. Akad. Nauk SSSR 283, 1060-1064 (1985; Zbl 0588.17015); Proc. Int. Congr. Math., Berkeley 1986, Vol. 1, 798-820 (1987; Zbl 0667.16003)] and \textit{M. Jimbo} [Lett. Math. Phys. 10, 63-69 (1985; Zbl 0587.17004); 11, 247-252 (1986; Zbl 0602.17005)], where \(q=e^ h\in {\mathbb{C}}\) in general and in this duality relation one has to restrict q,h\(\in {\mathbb{R}}\) and \(\mu =e^{-2h}\). Similarly, \textit{L. D. Faddeev}, \textit{N. Yu. Reshetikhin} and \textit{L. A. Takhtadzhan} [Quantization of Lie groups and Lie algebras, Algebraic Analysis 1, 129-139 (1989; Zbl 0677.17010)] and \textit{M. Rosso} [preprint Ecole Polytechnique 1989] constructed by duality from \(U_ q({\mathfrak g})\), with \({\mathfrak g}\) a Lie algebra of type \(B_ N\), \(C_ N\), \(D_ N\), the compact matrix quantum groups \(SO_{\mu}(N)\) and \(Sp_{\mu}(N).\) Let \(G_ h\) be any of the matrix quantum groups \(SU_{\mu}(N+1)\), \(SO_{\mu}(N)\) or \(Sp_{\mu}(N)\), let \({\mathcal A}_ h\) be the *-Hopf algebra of representative elements corresponding to \(G_ h\). The author shows that the algebra \({\mathcal A}_ h\) is isomorphic as a co-algebra to the algebra \({\mathcal A}_ 0\) (i.e., to the representative functions on the classical group). Thus \({\mathcal A}_ h\) may be identified with \({\mathcal A}_ 0\), however, with an associative product, called the star-product, which is a deformation of the original commutative product of \({\mathcal A}_ 0\), connected to the Fourier transform in a manner similar to the construction of quantum mechanics from classical mechanics in phase space. In fact the map \(W_ h: {\mathcal A}_ 0\to {\mathcal A}_ h\) which realizes the above isomorphism is an analog of the Weyl correspondence and is called the Weyl transformation.
scientific article; zbMATH DE number 4198264