[Paper Review] Projective Linear Monoids and Hinges

Mutsumi Saito|arXiv (Cornell University)|Nov 4, 2017

Advanced Topics in Algebra2 references3 citations

TL;DR

This paper introduces a new compactification, ℙM(V), of the projective linear group PGL(V) that acts naturally on the projective space ℙ(V). By constructing a monoid structure compatible with mappings on ℙ(V), the author proves ℙM(V) is compact and establishes a surjective continuous map from an open dense subset of ℙM(V) to Neretin's hinge compactification Hinge*(V), revealing a deep structural link between the two compactifications.

ABSTRACT

Let V be a complex vector space. We propose a compactification PM(V) of the projective linear group PGL(V), which can act on the projective space P(V). After proving some properties of PM(V), we consider its relation to Neretin's compactification Hinge*(V).

Motivation & Objective

• To construct a compactification ℙM(V) of PGL(V) that preserves the natural action on ℙ(V), addressing limitations of existing compactifications.
• To define a monoid structure on ℙM(V) compatible with the monoid of self-maps on ℙ(V), ensuring consistent action.
• To establish a topological and algebraic relationship between ℙM(V) and Neretin’s Hinge*(V), a known compactification of PGL(V).
• To prove that ℙM(V) is compact using net convergence, providing a new topological framework for symmetric space compactifications.

Proposed method

• Define M(V) as sequences of nonzero linear maps between nested kernels in a complex vector space V, forming a monoid-like structure.
• Projectivize M(V) to obtain ℙM(V), using the projective space of homomorphisms to define a topology.
• Introduce a topology on ℙM(V) via neighborhoods defined by projective images of homomorphisms on intermediate kernel subspaces.
• Use nets and convergence arguments to prove that every maximal net in ℙM(V) converges, establishing compactness (Theorem 5.3).
• Construct a natural monoid homomorphism Φ: ℙM(V) → Map(ℙ(V)) that realizes the action of ℙM(V) on ℙ(V).
• Define a map φ: ℙM(V) → Hinge*(V) via wedge power maps and show its restriction to an open dense subset ℙM_H(V) is continuous and surjective.

Experimental results

Research questions

• RQ1 Can a compactification of PGL(V) be constructed that preserves the natural action on ℙ(V), unlike previous compactifications?
• RQ2 How does the monoid structure of ℙM(V) relate to the monoid of endomorphisms on ℙ(V)?
• RQ3 What is the topological and algebraic relationship between ℙM(V) and Neretin’s Hinge*(V)?
• RQ4 Is ℙM(V) compact, and can this be proven via net convergence in a non-metric setting?

Key findings

• ℙM(V) is compact, proven via the convergence of every maximal net in the space (Theorem 5.3).
• The natural map Φ: ℙM(V) → Map(ℙ(V)) is a monoid homomorphism, confirming that ℙM(V) acts on ℙ(V).
• There exists a surjective continuous map from an open dense subset ℙM_H(V) of ℙM(V) to Neretin’s Hinge*(V), establishing a topological link.
• The monoid structure of ℙM(V) is compatible with the monoid of self-maps on ℙ(V), while that of a variant of Hinge*(V) is compatible with ∏_{i=1}^{n-1} End(∧^i V), showing distinct algebraic behaviors.
• The map λ̄: ℙM(V) → ∏_{k=1}^{n-1} ℙEnd(∧^k V) is continuous, and its composition with Neretin’s embedding λ° gives the surjective map to Hinge*(V).
• In Example 8.4, the limit of ∧^3 A(ε) as ε→0 converges to the restriction of ∧^3 id_V on a specific subspace, confirming compatibility with ℙM(V)’s construction.