Geometry of Weighted Homogeneous Spaces
Mohammad Reza Rahmati, Gerardo Flores

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 where is weight function defined on the set of simple roots of , by an element in the highest Weyl chamber. The weight function describes the action of the maximal torus on different Bruhat cells and is well behaved via the change of coordinates defined by the action of the Weyl group . 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 as a whole compact quotient of 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…
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic structures and combinatorial models · Advanced Topics in Algebra
