Discrete Integrable Systems, Positivity, and Continued Fraction Rearrangements

作者：Di Francesco Philippe^*

来源：Letters in Mathematical Physics, 2011, 96(1-3): 299-324.

DOI：10.1007/s11005-010-0429-x

• cluster algebras
• Laurent phenomenon
• positivity
• integrable systems
• non-commutative
• continued fractions

In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the Q- and T -systems based on A (r) . The initial data of the systems are seen as cluster variables in a suitable cluster algebra, and may evolve by local mutations. We show that the solutions are always expressed as Laurent polynomials of the initial data with non-negative integer coefficients. This is done by reformulating the mutations of initial data as local rearrangements of continued fractions generating some particular solutions, that preserve manifest positivity. We also show how these techniques apply as well to non-commutative settings.

• 出版日期2011-6
• 单位中国地震局