2000/11/23 by Yuri I. Manin, Manin, Yuri I.
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG
paper · pdf · doi:10.48550/arxiv.math/0011197
26 pp., amstex file, no figures
arxiv created 2000/11/23 · arxiv updated 2009/11/30
A linear algebraic group G is represented by the linear space of its algebraic functions F(G) endowed with multiplication and comultiplication which turn it into a Hopf algebra. Supplying G with a Poisson structure, we get a quantized version Fq(G) which has the same linear structure and comultiplication, but deformed multiplication. This paper develops a similar theory for abelian varieties. A description of abelian varieties A in terms of linear algebra data was given by Mumford: F(G) is replaced by the graded ring of theta functions with symmetric automorphy factors, and comultiplication is replaced by the Mumford morphism M^* acting on pairs of points as M(x,y)=M(x+y,x-y). After supplementing this by a Poisson structure and replacing the classical theta functions by the quantized ones, introduced by the author earlier, we obtain a structure which essentially coincides with the classical one so far as comultiplication is concerned, but has a deformed multiplication which moreover becomes only partial. The classical graded ring is thus replaced by a linear category. Another important difference from the linear case is that abelian varieties with different period groups (for multiplication) and different quantization parameters (for comultiplication) become interconnected after quantization.