[Paper Review] Positive structures in Lie theory

G. Lusztig|arXiv (Cornell University)|Dec 21, 2018

Algebraic structures and combinatorial models7 references4 citations

TL;DR

This paper introduces a unified framework for positive structures in Lie theory by defining the totally positive (TP) monoid $G(\mathbb{R}_{>0})$ and its upper triangular part $U^+(\mathbb{R}_{>0})$ via generators and relations over a semifield, independent of the ambient Lie group. The key contribution is that this TP structure serves as the foundational form from which all Chevalley groups $G(\mathbb{k})$ over arbitrary fields $\mathbb{k}$ can be reconstructed via rational relations with integer coefficients, revealing a deep algebraic-geometric duality in Lie theory.

ABSTRACT

We discuss the non-negative monoid attached to a Cartan matrix following my 1994 paper.

Motivation & Objective

• To establish a new, intrinsic definition of the totally positive monoid $G(\mathbb{R}_{>0})$ for a semisimple Lie group using generators and relations, independent of the ambient Lie group $G(\mathbb{R})$.
• To show that the defining relations of $G(\mathbb{R}_{>0})$ are rational functions with integer coefficients, enabling their extension to any field $\mathbb{k}$ to reconstruct Chevalley groups $G(\mathbb{k})$.
• To demonstrate that the TP monoid over $\mathbb{R}_{>0}$ serves as the universal 'birational form' of a semisimple group, from which all other group forms are derived.
• To unify diverse developments in Lie theory—such as cluster algebras, higher Teichmüller theory, and quantum groups—under a common positive structure framework.

Proposed method

• Define a positive structure on a set $X$ via a family of bijections $f_j: K^m \to X$ from a semifield $K$, where $K$ is $\mathbb{R}_{>0}$, $\mathbb{R}(t)_{>0}$, $\mathbb{Z}$, or $\{1\}$, with compatible operations.
• Construct the monoid $U^+(\mathbb{R}_{>0})$ as the image of a positive structure on the unipotent radical, using generators corresponding to positive root spaces.
• Define $G(\mathbb{R}_{>0})$ via generators and relations derived from the canonical basis of quantum groups, ensuring positivity and integrality of relations.
• Use the semifield homomorphism $\alpha: \mathbb{R}(t)_{>0} \to \mathbb{Z}$ to connect geometric positive structures with piecewise-linear (tropical) objects, establishing an early form of tropicalization in Lie theory.
• Reconstruct the Chevalley group $G(\mathbb{k})$ over any field $\mathbb{k}_0$ by reducing the relations of $G(\mathbb{R}_{>0})$ modulo $\mathbb{Z} \to \mathbb{k}_0$, yielding a birational model of the group.
• Define the non-negative part of the flag manifold $\mathcal{B}(K)$ as the intersection of the flag manifold with lines having non-negative coordinates in the canonical basis, and show that $G(K)$ acts on it via positivity.

Experimental results

Research questions

• RQ1 How can the totally positive monoid $G(\mathbb{R}_{>0})$ be defined independently of the ambient Lie group $G(\mathbb{R})$, using only generators and relations?
• RQ2 What is the role of the canonical basis in quantum groups in ensuring the positivity and integrality of the relations defining $G(\mathbb{R}_{>0})$?
• RQ3 How can the relations defining $G(\mathbb{R}_{>0})$ be reduced to define a group over an arbitrary field $\mathbb{k}_0$, thereby reconstructing the Chevalley group $G(\mathbb{k}_0)$?
• RQ4 In what sense is the TP monoid over $\mathbb{R}_{>0}$ the 'most basic' form of a semisimple group, from which all other forms are derived?
• RQ5 How does the action of $G(\mathbb{R}_{>0})$ on the non-negative part of the flag manifold $\mathcal{B}(K)$ reflect the positivity of the canonical basis?

Key findings

• The TP monoid $G(\mathbb{R}_{>0})$ is defined via generators and relations that are rational functions with integer coefficients, making it simpler than the definition of $G(\mathbb{R})$.
• The defining relations of $G(\mathbb{R}_{>0})$ are independent of the ambient Lie group and can be reduced modulo $\mathbb{Z} \to \mathbb{k}_0$ to yield a birational model of the Chevalley group $G(\mathbb{k}_0)$.
• The Chevalley group $G(\mathbb{k}_0)$ is realized as a subgroup of the automorphism group of the quotient field of the coordinate ring of $\mathbb{k}^M$, constructed from the relations of $G(\mathbb{R}_{>0})$.
• The non-negative part of the flag manifold $\mathcal{B}(K)$ is preserved under the action of $G(K)$, due to the positivity of the canonical basis in integrable highest weight representations.
• The semifield homomorphism $\alpha: \mathbb{R}(t)_{>0} \to \mathbb{Z}$ provides a bridge between geometric positive structures and piecewise-linear combinatorics, establishing an early instance of tropicalization in Lie theory.
• The construction shows that the TP monoid over $\mathbb{R}_{>0}$ is the universal form from which all other group forms over fields are derived, via reduction of relations with integer coefficients.