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Glorious pairs of roots and Abelian ideals of a Borel subalgebra
Journal of Algebraic Combinatorics ( IF 0.6 ) Pub Date : 2019-11-04 , DOI: 10.1007/s10801-019-00911-9
Dmitri I. Panyushev

Let \({{\mathfrak {g}}}\) be a simple Lie algebra with a Borel subalgebra \({{\mathfrak {b}}}\). Let \(\Delta ^+\) be the corresponding (po)set of positive roots and \(\theta \) the highest root. A pair \(\{\eta ,\eta '\}\subset \Delta ^+\) is said to be glorious, if \(\eta ,\eta '\) are incomparable and \(\eta +\eta '=\theta \). Using the theory of abelian ideals of \({{\mathfrak {b}}}\), we (1) establish a relationship of \(\eta ,\eta '\) to certain abelian ideals associated with long simple roots, (2) provide a natural bijection between the glorious pairs and the pairs of adjacent long simple roots (i.e., some edges of the Dynkin diagram), and (3) point out a simple transform connecting two glorious pairs corresponding to the incident edges in the Dynkin diagram. In types \({{\mathbf {\mathsf{{{DE}}}}}}_{}\), we prove that if \(\{\eta ,\eta '\}\) corresponds to the edge through the branching node of the Dynkin diagram, then the meet \(\eta \wedge \eta '\) is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type \({{\mathbf {\mathsf{{{A}}}}}}_{}\). As an application, we describe the minimal non-abelian ideals of \({{\mathfrak {b}}}\).



中文翻译:

Borel次代数的光荣成对和Abelian理想

\({{\ mathfrak {g}}} \)是具有Borel子代数\({{\ mathfrak {b}}} \)的简单李代数。令\(\ Delta ^ + \)是相应的正根(po)集,而\(\ theta \)是最高根。一对\(\ {\ ETA,\ ETA“\} \子集\德尔塔^ + \)被说成是光荣,如果\(\ ETA,\ ETA '\)所无法比拟的和\(\ ETA + \ ETA' = \ theta \)。使用\({{\ mathfrak {b}}} \)的阿贝尔理想理论,我们(1)建立\(\ eta,\ eta'\)的关系对于某些与长简单根相关的阿贝尔理想,(2)在光荣对和相邻的长简单根对之间提供自然双射(即,Dynkin图的某些边),并且(3)指出一个简单的变换连接对应于Dynkin图中入射边缘的两个光荣对。在类型\({{\ mathbf {\ mathsf {{{DE}}}}} _ {} \)中,我们证明如果\(\ {\ eta,\ eta'\} \)对应于Dynkin图的分支节点,则Meet \(\ eta \ wedge \ eta'\)是唯一的最大非交换根。除类型\({{\ mathbf {\ mathsf {{{{A}}}}}} __ {} \)之外,所有其他类型都具有此属性的类似物。作为应用程序,我们描述\({{\ mathfrak {b}}} \)的最小非阿贝尔理想。

更新日期:2019-11-04
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