On the automorphisms of the Drinfel'd double of a Borel Lie subalgebra

2020/02/09 by Michaël Bulois, Bulois, Michaël, Nicolas Ressayre +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.RT #msc:17B45 #msc:17B67

paper · pdf · doi:10.48550/arxiv.2002.03395

17 pages, 1 table. It was asserted in second version that an automorphism of a Kac-Moody algebra associated to an automorphism of the extended Dynkin diagram is C[t]-linear. In fact, we found no proof for this, but we provide some evidence in this direction

arxiv created 2021/06/18 · arxiv updated 2021/06/21

Abstract

Let \mathfrak g be a complex simple Lie algebra with Borel subalgebra \mathfrak b. Consider the semidirect product I\mathfrak b=\mathfrak b\ltimes\mathfrak b^*, where the dual \mathfrak b^* of \mathfrak b, is equipped with the coadjoint action of \mathfrak b and is considered as an abelian ideal of I\mathfrak b. We describe the automorphism group Aut(I\mathfrak b) of the Lie algebra I\mathfrak b. In particular we prove that it contains the automorphism group of the extended Dynkin diagram of \mathfrak b. In type A_n, the dihedral subgroup was recently proved to be contained in Aut(I\mathfrak b) by Dror Bar-Natan and Roland Van Der Veen in arXiv:2002.00697 (where I\mathfrak b is denoted by I\mathfrak u_n). Their construction is handmade and they ask for an explanation: this note fully answers the question.

Citations

• An Unexpected Cyclic Symmetry of I\mathfraku_n

Cited by

• Type A algebraic coherence conjecture of Pappas and Rapoport

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