论文标题
Toda系统的Takiff代数
Toda systems for Takiff algebras
论文作者
论文摘要
我们研究了附属于Tabiff代数$ \ Mathfrak {g} _n $的完全可集成的系统，扩展了开放的toda Systems of Split Split Simple Lie代数$ \ Mathfrak {G} $。关于coadexhindexhindexhoint Orbits $ \ Mathcal {O} $的Darboux坐标，汉密尔顿人的潜力是多项式和指数函数的产物。 $ \ mathfrak {g} _n $的运动方程方程的通用解决方案是使用称为喷气转换的微分运算符获得的。这些结果适用于基于$ \ mathfrak {sl}（2）$的$ 3 $ body问题，并将其扩展到$ a_ \ infty $的Soliton Solutions到相关的Takiff代数。然后将新的经典集成系统提升到通勤操作员的家族中，以封闭的代数解决，解决Vinberg问题，并在$ \ Mathcal {O} $上量化功能的泊松代数。
We study completely integrable systems attached to Takiff algebras $\mathfrak{g}_N$, extending open Toda systems of split simple Lie algebras $\mathfrak{g}$. With respect to Darboux coordinates on coadjoint orbits $\mathcal{O}$, the potentials of the hamiltonians are products of polynomial and exponential functions. General solutions for equations of motion for $\mathfrak{g}_N$ are obtained using differential operators called jet transformations. These results are applied to a $3$-body problem based on $\mathfrak{sl}(2)$, and to an extension of soliton solutions for $A_\infty$ to associated Takiff algebras. The new classical integrable systems are then lifted to families of commuting operators in an enveloping algebra, solving a Vinberg problem and quantizing the Poisson algebra of functions on $\mathcal{O}$.
