论文标题
仿射lie的群集变量 - 波森系统
Cluster variables for affine Lie--Poisson systems
论文作者
论文摘要
我们表明，在圆盘上拥有任何平面（环状或无环）网络，其唯一条件是所有$ n_1+m $源都与所有$ n_2+m $ sinks分开，我们可以构建一个集群的代数实现extine lie-poisson lie-poisson elgebra的元素$ r（λ，μ）t^{1}（λ）t^{2}（μ）= t^{2}（μ）t^{1}（λ）r（λ）在满足某些可逆性条件的情况下，我们可以将这种构造扩展到对量子回路代数的实现。具有量子回路代数，我们还可以构建扭曲的仰光代数或量子反射方程的实现。因此，每个这样的平面网络都对应于相应的无限维代数的符号叶。
We show that having any planar (cyclic or acyclic) directed network on a disc with the only condition that all $n_1+m$ sources are separated from all $n_2+m$ sinks, we can construct a cluster-algebra realization of elements of an affine Lie--Poisson algebra $R(λ,μ)T^{1}(λ)T^{2}(μ)=T^{2}(μ)T^{1}(λ)R(λ,μ)$ with $(n_1\times n_2)$-matrices $T(λ)$ corresponding to a planar directed network on an annulus. Upon satisfaction of some invertibility conditions, we can extend this construction to realizations of a quantum loop algebra. Having the quantum loop algebra we can also construct a realization of the twisted Yangian algebra, or that of the quantum reflection equation. Every such planar network therefore corresponds to a symplectic leaf of the corresponding infinite-dimensional algebra.
