The classification of differential structures on quantum 2-spheres
Based mainly on the following previous works [\textit{P. Podleś}, Lett. Math. Phys. 18, 107-119 (1989; Zbl 0702.53073), RIMS Kokyuroku 751 (1991); \textit{P. Podleś} and \textit{S. L. Woronowicz}, Commun. Math. Phys. 130, 381-431 (1990; Zbl 0703.22018); \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), Invent. Math. 93, 35-76 (1988; Zbl 0664.58044), Commun. Math. Phys. 122, 125-170 (1989; Zbl 0751.58042)] the author presents in this article the complete proof of the following theorem: For \(q\in[-1,1]/\{0\}\), \(c=0\) there exists a unique \({2\choose \cdot}\)- dimensional exterior algebra on the quantum sphere \(S^ 2_{qc}\) invariant with respect to the *-homomorphism \(\sigma_{qc}\) which describes the action of \(SU_ q(2)\) on \(S^ 2_{qc}\). For \(q\in(- 1,1)/\{0\}\), \(c\in(0,\infty]\) there are no \({2 \choose \cdot}\)- dimensional exterior algebras on \(S^ 2_{qc}\) invariant with respect to \(\sigma_{qc}\); which clearly classifies all quantum generalizations of the exterior algebra of differential forms on the standard 2-sphere \(S^ 2\). In the last section, the generalized directional derivatives corresponding to the exterior algebras on the quantum spheres \(S^ 2_{q0}\), \(q\in(-1,1)/\{0\}\), are investigated and the classical limits, \(q\to 0\), of these derivatives are provided.
scientific article; zbMATH DE number 165651