$q$-Analog of Gelfand–Graev Basis for the Noncompact Quantum Algebra $U_q(u(n,1))$

Raisa M. Asherova ; Čestmír Burdík ; Miloslav Havlíček ; Yuri F. Smirnov ; Valeriy N. Tolstoy

Symmetry, integrability and geometry: methods and applications, Tome 6 (2010)

Résumé

For the quantum algebra $U_q(\mathfrak{gl}(n+1))$ in its reduction on the subalgebra $U_q(\mathfrak{gl}(n))$ $Z_q(\mathfrak{gl}(n+1),\mathfrak{gl}(n))$ is given in terms of the generators and their defining relations. Using this $Z$-algebra we describe Hermitian irreducible representations of a discrete series for the noncompact quantum algebra $U_q(u(n,1))$ which is a real form of $U_q(\mathfrak{gl}(n+1))$, namely, an orthonormal Gelfand–Graev basis is constructed in an explicit form.

Détail

Keywords: quantum algebra; extremal projector; reduction algebra; Shapovalov form; noncompact quantum algebra; discrete series of representations; Gelfand–Graev basis.

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