2001/10/16 by M. V. Karasev, Karasev, M. V., E. M. Novikova +1
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #math.QA
paper · pdf · doi:10.48550/arxiv.math/0110164
21 pages, Amstex-file
arxiv created 2001/10/16 · openalex publication_date 2001/10/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a class of algebras with non-Lie commutation relations whose symplectic leaves are surfaces of revolution: a cylinder or a torus. Over each of such surfaces we introduce a family of complex structures and Hilbert spaces of antiholomorphic sections in which the irreducible Hermitian representations of the original algebra are realized. The reproducing kernels of these spaces are expressed in terms of the Riemann theta-function and its modifications. They generate quantum Kähler structures on the surface and the corresponding quantum reproducing measures. We construct coherent transforms intertwining abstract representations of an algebra with irreducible representations, and these transforms are also expressed via the theta-function.