论文标题
关于Fomin-Kirillov代数的维度和相关代数
On the dimension of the Fomin-Kirillov algebra and related algebras
论文作者
论文摘要
1999年，Fomin-Kirillov在发电机和关系方面介绍了二次代数$ \ MATHCAL {E} _M $，这是由分隔差的操作员$ \ partial_ {ij} $在polynomial Ring $ \ mathbffff pationan oct的差异操作员生成的代数的普遍二次封面。这些代数非常重要，因为它们与舒伯特的演算和几何形状以及量子基团和尼科尔代数的一般框架。 Fomin和Kirillov询问了$ \ Mathcal {E} _M $的维度。在本文中，我们证明$ \ Mathcal {e} _m $对于所有$ M \ geq 6 $都是无限的维度，这是一个众所周知的猜想。我们使用的技术依赖于Liu和Bazlov开发的编织差分微积分以及Sweedler介绍的Hopf代数的积分概念。
In 1999, Fomin-Kirillov introduced the quadratic algebras $\mathcal{E}_m$ in terms of generators and relations which are the universal quadratic cover of the algebra generated by divided difference operators $\partial_{ij}$ acting on the polynomial ring $\mathbf{k}[x_1,\ldots,x_m]$. These algebras are mostly important due to their relations to Schubert calculus and geometry and to the general framework of quantum groups and Nichols algebras. Fomin and Kirillov asked about the dimension of $\mathcal{E}_m$. In this paper, we prove that $\mathcal{E}_m$ is infinite dimensional for all $m\geq 6$ which was a well-known conjecture. The techniques we use rely on braided differential calculus as developed by Liu and Bazlov as well as on the notion of integrals for Hopf algebras as introduced by Sweedler.
