An extension of the Borel-Weil construction to the quantum group \(U_ q(n)\)
The well-known Borel-Weil theorem allows us to construct unitary irreducible representations of a compact Lie group realizing them on holomorphic sections of homogeneous holomorphic line bundles over Kähler manifolds. The aim of this paper is to give the extension of this construction for the quantum group \(U_ q(n)\). The authors employ a recursive approach which differs from the Borel-Weil procedure in that the induction is from an irreducible representation of the quantum subgroup \(U_ q(n-1)\times U_ q(1)\) rather than from a character of the maximal torus \(T_ q\). The base manifold is \(U_ q(n)/(U_ q(n- 1)\times U_ q(1))\) which has a complex structure. The recursive procedure assumes a prior construction of irreducible representations of \(U_ q(n-1)\) by the same method. The extension of the Borel-Weil construction to \(U_ q(n)\) differs in still another way from the procedure actually used in the classical case in that the authors use an analytic method rather than the purely geometric method.
scientific article; zbMATH DE number 90904
scientific article