Periodicities of Tsystems and Ysystems
Abstract
The unrestricted Tsystem is a family of relations in the Grothendieck ring of the category of the finitedimensional modules of the Yangian or the quantum affine algebra associated with a complex simple Lie algebra. The unrestricted Tsystem admits a reduction called the restricted Tsystem. In this paper we formulate the periodicity conjecture for the restricted Tsystems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Ysystems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).
 Publication:

arXiv eprints
 Pub Date:
 December 2008
 arXiv:
 arXiv:0812.0667
 Bibcode:
 2008arXiv0812.0667I
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Representation Theory
 EPrint:
 83 pages with 8 figures