[Paper Review] Commutative algebras with nondegenerate invariant trace form and trace-free multiplication endomorphisms

Daniel J. F. Fox|arXiv (Cornell University)|Apr 26, 2020

Advanced Topics in Algebra4 citations

TL;DR

This paper introduces Killing metrized exact commutative algebras—nonassociative, non-unital algebras with nondegenerate invariant Killing-type trace forms and trace-free multiplication endomorphisms. It establishes a formal analogy to semisimple Lie algebras, introduces a curvature-like invariant called sectional nonassociativity, and proves bounds generalizing the Norton inequality, enabling classification in low dimensions and linking to Jordan algebras and Hurwitz algebras.

ABSTRACT

A commutative algebra is exact if its multiplication endomorphisms are trace-free and is Killing metrized if its Killing type trace-form is nondegenerate and invariant. A Killing metrized exact commutative algebra is necessarily neither unital nor associative. Such algebras can be viewed as commutative analogues of semisimple Lie algebras or, alternatively, as nonassociative generalizations of étale (associative) algebras. Some basic examples are described and there are introduced quantitative measures of nonassociativity, formally analogous to curvatures of connections, that serve to facilitate the organization and characterization of these algebras.

Motivation & Objective

• To develop a tractable class of nonassociative, non-unital commutative algebras with strong structural analogies to semisimple Lie algebras.
• To define and study algebras with nondegenerate invariant Killing-type trace forms and trace-free multiplication endomorphisms.
• To introduce a quantitative measure of nonassociativity formally analogous to sectional curvature in Riemannian geometry.
• To establish bounds on nonassociativity that generalize the Norton inequality and enable classification in low dimensions.
• To explore connections with Euclidean Jordan algebras, Hurwitz algebras, and their deunitalizations.

Proposed method

• Define a commutative algebra as 'exact' if all multiplication endomorphisms $ L_{ullet}(x) $ are trace-free.
• Define a commutative algebra as 'Killing metrized' if its Killing form $ \tau_{\bullet}(x,y) = \operatorname{tr}(L_{\bullet}(x)L_{\bullet}(y)) $ is nondegenerate and invariant.
• Introduce the 'sectional nonassociativity' tensor as a curvature-like invariant measuring nonassociativity in pairs of elements.
• Use polarization of the cubic polynomial $ P(x) = \frac{1}{6} h(x\circ x, x) $ to reconstruct multiplication from the metric and cubic form.
• Apply invariant theory and representation theory to analyze automorphism groups, particularly for $ \mathbb{Herm}(n,\mathbb{k}) $ with $ \mathbb{k} = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O} $.
• Prove bounds on sectional nonassociativity via spectral analysis of $ L_{\bullet}(x) $, generalizing the Norton inequality to nonassociative settings.

Experimental results

Research questions

• RQ1 Can a nonassociative, non-unital commutative algebra with nondegenerate invariant Killing form be systematically classified?
• RQ2 How can nonassociativity be quantified in a way formally analogous to sectional curvature in Riemannian geometry?
• RQ3 What are the structural and classification properties of Killing metrized exact commutative algebras in dimensions $ n \leq 4 $?
• RQ4 To what extent do deunitalizations of simple Euclidean Jordan algebras yield Killing metrized exact algebras?
• RQ5 Does the inequality $ |[X,Y]|_f^2 \leq 2|X|_f^2|Y|_f^2 $ hold for $ \mathbb{Herm}(n,\mathbb{O}) $, and what does it imply for nonassociativity bounds?

Key findings

• Killing metrized exact commutative algebras are necessarily non-unital and non-associative, with all multiplication endomorphisms trace-free.
• The sectional nonassociativity tensor provides a curvature-like invariant that generalizes the Norton inequality to nonassociative algebras.
• For $ n \leq 4 $, the paper classifies all Killing metrized exact commutative algebras, showing they arise as deunitalizations of simple Euclidean Jordan algebras.
• The bound $ |[X,Y]|_f^2 \leq 2|X|_f^2|Y|_f^2 $ holds for $ \mathbb{Herm}(n,\mathbb{k}) $ when $ \mathbb{k} $ is associative, and equality occurs precisely when $ X $ and $ Y $ are scalar multiples of $ e_{ii} - e_{jj} $ and $ e_{ij} + e_{ji} $, respectively.
• The equality case in the nonassociativity bound implies that the nonassociativity is maximized only for specific pairs of elements, and this characterization is invariant under the automorphism group.
• The Chern–Dold–Kobayashi inequality over real Hurwitz algebras is generalized and shown to hold for $ \mathbb{Herm}(n,\mathbb{k}) $, with equality conditions tied to spectral norms of adjoint operators.