Wonderful compactification of a Cartan subalgebra of a semisimple Lie algebra

Li Yu (University of Chicago)

03-Dec-2020, 19:50-20:50 (3 years ago)

Abstract: Let $H$ be a Cartan subgroup of a semisimple algebraic group $G$ over the complex numbers. The wonderful compactification $\overline{H}$ of $H$ was introduced and studied by De Concini and Procesi. For the Lie algebra $\mathfrak{h}$ of $H$, we define an analogous compactification $\overline{\mathfrak{h}}$ of $\mathfrak{h}$, to be referred to as the wonderful compactification of $\mathfrak{h}$. The wonderful compactification of $\mathfrak{h}$ is an example of an "additive toric variety". We establish a bijection between the set of irreducible components of the boundary $\overline{\mathfrak{h}} - \mathfrak{h}$ of $\mathfrak{h}$ and the set of maximal closed root subsystems of the root system for $(G, H)$ of rank $r - 1,$ where $r$ is the dimension of $\mathfrak{h}$. An algorithm based on Borel-de Siebenthal theory that constructs all such root subsystems is given. We prove that each irreducible component of $\overline{\mathfrak{h}}- \mathfrak{h}$ is isomorphic to the wonderful compactification of a Lie subalgebra of $\mathfrak{h}$ and is of dimension $r - 1$. In particular, the boundary $\overline{\mathfrak{h}} - \mathfrak{h}$ is equidimensional. We describe a large subset of the regular locus of $\overline{\mathfrak{h}}$. As a consequence, we prove that $\overline{\mathfrak{h}}$ is a normal variety.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic


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