Theta functions, quantum tori and Heisenberg groups
| dc.creator | Manin, Yuri I. | |
| dc.date | 2000-11-23 | |
| dc.date.accessioned | 2026-07-07T04:38:48Z | |
| dc.date.available | 2026-07-07T04:38:48Z | |
| dc.description | 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 $F_q(G)$ 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. | |
| dc.description | 26 pp., amstex file, no figures | |
| dc.identifier | https://arxiv.org/abs/math/0011197 | |
| dc.identifier | http://arxiv.org/abs/math/0011197 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/60422 | |
| dc.subject | Algebraic Geometry | |
| dc.title | Theta functions, quantum tori and Heisenberg groups | |
| dc.type | text |