Differential calculus on quantum homogeneous spaces
The quantum tangent space of a covariant first-order differential calculus (FODC) over a quantum homogeneous space is established, and the generators of FODC over the Podleś' quantum sphere \(C_q [\mathbb{S}_c^2]\) are determined as an application. An FODC over \({\mathcal B}\), an algebra over \(\mathbb{C}\) is a \({\mathcal B}\)-bimodule \(\Gamma\) together with a \(\mathbb{C}\)-linear map d\(: {\mathcal B}\to\Gamma\), where \[ \Gamma=\text{Lin}_\mathbb{C}\{adb c\mid a.b.c\in {\mathcal B}\},\quad \text{d}ab=\text{d}a b+ a\text{d}b, \] [cd. \textit{A. Klimyk} and \textit{K. Schmüdgen}, ``Quantum Groups and their Representations'', Springer, Heidelberg (1997; Zbl 0891.17010)]. Let \({\mathcal A}\) be a Hopf algebra and \(\Delta_{\mathcal B}:{\mathcal B} \to {\mathcal A}\otimes {\mathcal B}\) a left comodule structure on \({\mathcal B}\). \(\Gamma\) is called left-covariant if \(\Gamma\) possesses a left \({\mathcal A}\)-comodule structure \(\Delta_\Gamma:\Gamma\to{\mathcal A}\otimes\Gamma\) such that \[ \Delta_\Gamma (a\text{d}b c)= (\Delta_{\mathcal B}a)(\text{Id} \otimes\text{d})\Delta_{\mathcal B} b)(\Delta_{{\mathcal B}}{\mathcal C}). \] Let \({\mathcal A}=U^0_{\mathcal C}\) be the dual Hopf algebra generated by the matrix coefficients of all \(U\)-modules in \({\mathcal C}\), where \(U\) is a Hopf algebra over \(\mathbb{C}\) with bijective antipode \({\mathcal S}\) and a right coideal subalgebra \(K\), and \({\mathcal C}\) is a tensor category of finite-dimensional left \(U\)-modules (Sect.2). If \({\mathcal A}\) separates the elements of \(U\) and the antipode is bijective, then the quantum homogeneous space is the left coideal subalgebra \[ {\mathcal B}=\bigl\{b\in {\mathcal A}\mid b_{(1)}b_{(2)}(k)= 0,\;k\in K^+\bigr\} K^+= \bigl\{k-\varepsilon (k)\mid k\in K\bigr\} \] (Sect. 3), cf. \textit{E. F. Müller} and \textit{H. Schneider} [Isr. J. Math. 111, 157-190 (1999; Zbl 1001.17015)]. Let \({\mathcal B}^+= \{b\in{\mathcal B}\mid \varepsilon (b)=0\}\) and \[ {\mathcal R}=\left\{ \sum_i\varepsilon (a_i)b_i^+ \mid\sum_i a_i\text{d}b_i =0\right\} \subset {\mathcal B}^+,\quad b;=b- \varepsilon(b). \] Then the quantum tangent space \(T^+_{\mathcal R}\) of \(\Gamma\) is defined to be \[ T^+_{\mathcal R}=\bigl\{ t\in T_{\mathcal R}\mid t(1)= 0\bigr\}\qquad T_{\mathcal R}=\bigl\{f \in{\mathcal B}^*\mid f(x)= 0,\;x\in{\mathcal R}\bigr\} \] (Sect. 4). In Sect. 5, it is shown that \(dW\), \(W \subset {\mathcal B}\) a left \({\mathcal A}\)-comodule, generates \(\Gamma\) as a left \({\mathcal B}\)-module if and only if the elements of \(W\) separate the quantum tangent space of \(\Gamma\) (Lemma 7). In Sect. 6, the induced FODC is discussed by using quantum tangent space. Then constructing Podleś quantum sphere as a left \(\mathbb{C}_q [SL(2)]\)-comodule, generators of FODC over Podleś' quantum sphere are determined in Sect. 7.
scientific article; zbMATH DE number 1967857