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On the dimension of the Fomin-Kirillov algebra and related algebras

Christoph Bärligea

Abstract

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.

On the dimension of the Fomin-Kirillov algebra and related algebras

Abstract

In 1999, Fomin-Kirillov introduced the quadratic algebras in terms of generators and relations which are the universal quadratic cover of the algebra generated by divided difference operators acting on the polynomial ring . 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 . In this paper, we prove that is infinite dimensional for all 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.

Paper Structure

This paper contains 5 sections, 4 theorems, 13 equations, 1 figure.

Table of Contents

  1. toc0Introduction
  2. toc0Coxeter groups
  3. toc0Disjoint systems in Coxeter groups
  4. toc0Nichols algebras and braided differential calculus
  5. Yetter-Drinfeld category over a Hopf algebra

Key Result

Theorem 1.1

The Fomin-Kirillov algebra $\EuScript{E}_m$ is infinite dimensional for all $m\geq 6$.

Figures (1)

  • Figure 1: For this figure, we assume that $W=\mathbb{S}_6$ and that the notation from Section \ref{['sec:coxeter']} is realized for $\mathbb{S}_6$. Let $w_1$ and $w_2$ be the elements of $\mathbb{S}_6$ as defined in Example \ref{['ex:S6']}. The barycentric graph illustrates all positive roots in $R^+$ where $\beta_1,\beta_2,\beta_3,\beta_4,\beta_5$ denote all simple roots in $\Delta$ with the labeling as in bourbaki_roots. The positive roots labeled with $1$ and $2$ correspond each up to multiplication with $w_o$ from the right to a permutation in $\mathbb{S}_6$ (in this case to $w_1,w_1w_o$ and $w_2,w_2w_o$, cf. Remark \ref{['rem:T']}) because they form a diagram with straight lines isomorphic to the Dynkin diagram of type $\mathsf{A}_5$ where we possibly allow reflection along the horizontal bottom line below the graph. They correspond to elements in the centralizer of the longest element of $\mathbb{S}_6$ because their diagrams are symmetric with respect to the vertical line in the middle of the graph. They correspond to permutations without rising or falling succession because none of them is simple. If we denote by $T_1,T_2$ the set of positive roots labeled with $1,2$, respectively, then we have $T_1=T_{w_1}$ and $T_2=T_{w_2}$.

Theorems & Definitions (26)

  • Theorem 1.1: Corollary \ref{['cor:main']}
  • Remark 1.2
  • proof : Setup of the paper
  • Theorem 1.3: Theorem \ref{['thm:main']}
  • Remark 1.4
  • proof : Strategy of the proof of Theorem \ref{['thm:inf_Bm-intro']}
  • proof : Intuition of the paper
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3: The center of $W$
  • ...and 16 more