Abelian Ideals and the Variety of Lagrangian Subalgebras

2020/10/09 by Sam Evens, Yu Li, Evens, Sam +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2010.04358

openalex publication_date 2020/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a semisimple algebraic group G of adjoint type with Lie algebra \mathfrak g over the complex numbers, we establish a bijection between the set of closed orbits of the group G \ltimes \mathfrak g^∗ acting on the variety of Lagrangian subalgebras of \mathfrak g \ltimes \mathfrak g^∗ and the set of abelian ideals of a fixed Borel subalgebra of \mathfrak g. In particular, the number of such orbits equals 2^{rk \mathfrak g} by Peterson's theorem on abelian ideals.

Citations

• Abelian ideals in a Borel subalgebra of a complex simple Lie algebra

Related

• On the variety of Lagrangian subalgebras, II
• Adjoint orbits of semi-simple Lie groups and Lagrangean submanifolds
• On adjoint orbits in nilpotent ideals of a Borel subalgebra
• Structure of the coadjoint orbits of Lie groups
• Remarks on the abelian ideals of a Borel subalgebra