Coisotropic Subalgebras of Complex Semisimple Lie Bialgebras

TL;DR

This paper generalizes a method for constructing coisotropic subalgebras in complex semisimple Lie bialgebras by using Lagrangian subalgebras and torus fixed points, expanding on previous work by Marco Zambon.

Contribution

It introduces a new construction of coisotropic subalgebras based on Lagrangian subalgebras and torus fixed points, extending Zambon's original approach.

Findings

Construction based on Lagrangian subalgebras of the double $rak{g} igoplus rak{g}$

Identification of coisotropic subalgebras with torus fixed points

Generalization of Zambon's method for complex semisimple Lie bialgebras

Abstract

In his paper "A Construction for Coisotropic Subalgebras of Lie Bialgebras", Marco Zambon gave a way to use a long root of a complex semisimple Lie biaglebra $\mathfrak{g}$ to construct a coisotropic subalgebra of $\mathfrak{g}$. In this paper, we generalize Zambon's construction. Our construction is based on the theory of Lagrangian subalgebras of the double $\mathfrak{g}\oplus\mathfrak{g}$ of $\mathfrak{g}$, and our coisotropic subalgebras correspond to torus fixed points in the variety $\mathcal{L}(\mathfrak{g}\oplus\mathfrak{g})$ of Lagrangian subalgebras of $\mathfrak{g}\oplus\mathfrak{g}$.