Geometry of Weighted Homogeneous Spaces

TL;DR

This paper introduces weighted homogeneous spaces (WHS), exploring their algebraic, geometric, and combinatorial properties, including their structure, isomorphisms, and connections to cluster algebras and weighted projective spaces.

Contribution

It generalizes existing theories of homogeneous spaces by incorporating weight functions, providing criteria for isomorphisms, and linking WHS to weighted cluster algebras and projective embeddings.

Findings

WHS can be expressed as quotients of G/P by finite abelian groups.

Criteria for isomorphism between different WHS are established.

The coordinate ring of a WHS is a weighted cluster algebra of finite type.

Abstract

In this paper, we define the weighted homogeneous space (WHS), denoted by $\frac{G}{P}[\psi_H]$ where $\psi_H$ is weight function defined on the set of simple roots of $G$, by an element $H$ in the highest Weyl chamber. The weight function $\psi_H$ describes the action of the maximal torus $T$ on different Bruhat cells and is well behaved via the change of coordinates defined by the action of the Weyl group $W$. The major effort in this text is to prove basic algebraic and geometric properties of a weighted homogeneous space. The definition can be compared with an existing version given by Reid-Corti \cite{CR}. Additionally, we express $\frac{G}{P}[\psi_H]$ as a whole compact quotient of $G/P$ by a certain action of a finite abelian group. Besides, it is presented a criterion when two WHS with possibly different weight systems are isomorphic. The criteria give a simple method to understand the regular maps between two WHS's, defined by matrices with specific polynomial entries. We also explain invariant K\"ahler differentials on WHS by using certain potential functions on $G$. Our contribution is a generalization of the results presented in \cite{Al, AL, AKQ}. For that, we explain how the weights affect different computations of chern classes of line bundles given in \cite{AL, AKQ}. Finally, we provide a result on the coordinate ring of a WHS by cluster algebras associated to weighted quivers. Specifically, we show that the coordinate ring of a WHS is a weighted cluster algebra of finite type. In this case, the corresponding Dynkin quiver is equipped with a weight function defined on the vertices where the mutations also affect the weights. We present an embedding of a WHS in a product of weighted projective spaces, showing that the coordinate ring is a weighted graded algebra.