Theta functions, quantum tori and Heisenberg groups

A linear algebraic group G is represented by the linear space of its algebr aic functions F(G) endowed with multiplication and comultiplication which t urn it into a Hopf algebra. Supplying G with a Poisson structure, we get a quantized version F-q(G) which has the same linear structure and comultipli cation, but deformed multiplication. This paper develops a similar theory f or Abelian varieties. A description of Abelian varieties A in terms of line ar algebra data was given by Mumford: F(G) is replaced by the graded ring o f theta functions with symmetric automorphy factors, and comultiplication i s replaced by the Mumford morphism M* acting on pairs of points as M(x, y) = (x + y, x - y). After supplementing this with a Poisson structure and rep lacing the classical theta functions by the quantized theta functions, intr oduced earlier by the author, we obtain a structure which essentially coinc ides with the classical one so far as comultiplication is concerned, but ha s a deformed multiplication which, moreover, becomes only partial. The clas sical graded ring is thus replaced by a linear category. Another important difference from the linear case is that Abelian varieties with different pe riod groups (for multiplication) and different quantization parameters (for comultiplication) become interconnected after quantization.