块上三角双线性型Poisson代数与辫群action
Poisson algebras of block-upper-triangular bilinear forms and braid group action
摘要：
本文研究了$C^{N}$上双线性型空间上的二次Poisson代数结构，并证明了对任何$n,m\in N$，使得$n m =N$，Poisson代数对由大小为$m\times m$的块组成的块上三角矩阵的双线性型空间的限制是Poisson.对中心元进行了分类，并证明了李代数体结构与Poisson代数相容。我们整合这个代数胚，得到相应的块上三角双线性形式的态射广群。群胚元素自动保持泊松代数。然后，我们得到的辫子群作用的泊松代数内的广群的初等生成元。我们讨论的仿射和量化的泊松代数，表明在$m=1$的情况下，量子仿射代数是扭曲的$q$ -杨为${o}_n$和$m=2$是扭曲的$q$ -杨为${sp}_{2n}$。我们描述了这两个例子中的量子辫群作用，并推测了任何$m>2$的这种作用的形式。
英文摘要：
In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on $C^{N}$ with the property that for any $n,m\in N$ such that $n m =N$, the restriction of the Poisson algebra to the space of bilinear forms with block-upper-triangular matrix composed from blocks of size $m\times m$ is Poisson. We classify all central elements and characterise the Lie algebroid structure compatible with the Poisson algebra. We integrate this algebroid obtaining the corresponding groupoid of morphisms of block-upper-triangular bilinear forms. The groupoid elements automatically preserve the Poisson algebra. We then obtain the braid group action on the Poisson algebra as elementary generators within the groupoid. We discuss the affinisation and quantisation of this Poisson algebra, showing that in the case $m=1$ the quantum affine algebra is the twisted $q$-Yangian for ${o}_n$ and for $m=2$ is the twisted $q$-Yangian for ${sp}_{2n}$. We describe the quantum braid group action in these two examples and conjecture the form of this action for any $m>2$.
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