4 research outputs found
Multiple noncommutative tori and Hopf algebras
We derive the Kac-Paljutkin finite-dimensional Hopf algebras as finite
fibrations of the quantum double torus and generalize the construction for
quantum multiple tori.Comment: 18 pages; AMSLaTeX (major revision, the construction of dual
rewritten using approach of multiplier Hopf algebras, references added
Desargues maps and the Hirota-Miwa equation
We study the Desargues maps \phi:\ZZ^N\to\PP^M, which generate lattices
whose points are collinear with all their nearest (in positive directions)
neighbours. The multidimensional compatibility of the map is equivalent to the
Desargues theorem and its higher-dimensional generalizations. The nonlinear
counterpart of the map is the non-commutative (in general) Hirota--Miwa system.
In the commutative case of the complex field we apply the nonlocal
-dressing method to construct Desargues maps and the
corresponding solutions of the equation. In particular, we identify the
Fredholm determinant of the integral equation inverting the nonlocal
-dressing problem with the -function. Finally, we establish
equivalence between the Desargues maps and quadrilateral lattices provided we
take into consideration also their Laplace transforms.Comment: 17 pages, 5 figures; v2 - presentation improve