Quantum Stiefel manifold and double cosets of quantum unitary group (Q1306258)

The first part of the paper contains a concise summary of the basic definitions and properties concerning the Hopf algebras \(GL_q(n,\mathbb{C})\), \(U_q(gl(n,\mathbb{C}))\) and \(U_q(n)\), as well as the construction of quantum (left, right or double) cosets. To an epimorphism of Hopf algebras, \(\gamma:H_1\to H_2\), one relates the left and right coaction, \(L_{H_2}=(\gamma\otimes\text{id})\circ\Delta_1:H_1\to H_2\otimes H_1\) and \(R_{H_2}=(\text{id}\otimes\gamma)\circ\Delta_1:H_1\to H_1\otimes H_2\). Here \(\Delta_1\) is the comultiplication in \(H_1\). The sets \(H_2\backslash H_1\), \(H_1/H_2\) and \(H_2\backslash H_1/H_2\) are formed by left-, right- and bi-invariant elements in \(H_1\), respectively. The authors particularly concentrate on the quantum Stiefel manifolds \(S_q^{n,m}=U_q(n)/U_q(n-m)=SU_q(n)/SU_q(n-m)\). They explicitly describe the \(\ast\)-algebra \(S_q^{n,m}\) in terms of generators \(t_{ij}\), \(t_{ij}^\ast\). Further, using the known representation theory for \(SU_q(n)\) they classify the \(\ast\)-representations of \(S_q^{n,m}\). A formula for the invariant integral on \(S_q^{n,m}\) is derived as well. These results are then extended to the double coset \(U_q(n-m)\backslash U_q(n)/U_q(n-m)\) since it is in fact the intersection of \(U_q(n)/U_q(n-m)\) and \(U_q(n-m)\backslash U_q(n)\).