[Paper Review] Pentagon and hexagon equations

Hidekazu Furusho|arXiv (Cornell University)|Feb 6, 2007

Algebraic structures and combinatorial models5 references4 citations

TL;DR

This paper proves that Drinfel'd's pentagon equation implies his two hexagon equations in the contexts of Lie algebras, pro-unipotent groups, pro-$l$ groups, and pro-nilpotent groups. The key result is that the pentagon equation alone defines the associator set and the Grothendieck-Teichmüller group up to a quadratic extension, with the hexagon equations arising uniquely from the pentagon condition via a field extension involving the coefficient $ c_2(\varphi) $. The result shows that the hexagon equations are not independent relations but consequences of the pentagon equation in these settings.

ABSTRACT

The author will prove that Drinfel'd's pentagon equation implies his two hexagon equations in the Lie algebra, pro-unipotent, pro-$l$ and pro-nilpotent contexts.

Motivation & Objective

• To establish that Drinfel'd's pentagon equation implies the two hexagon equations in the Lie algebra, pro-unipotent, pro-$l$, and pro-nilpotent settings.
• To show that the set of associators and the Grothendieck-Teichmüller group are defined essentially by the pentagon equation alone, with hexagon equations being derivable.
• To clarify the logical dependency between the pentagon and hexagon axioms in the context of braided tensor categories and their deformations.
• To demonstrate that the parameter $ \mu $ or $ \lambda $, required for the hexagon equations, arises naturally from the coefficient $ c_2(\varphi) $ or $ c_2(f) $, necessitating a quadratic extension of the base field.

Proposed method

• Use of Drinfel'd's 'gadgets' to relate group-like elements in universal enveloping algebras to associators satisfying the pentagon equation.
• Construction of a group isomorphism between the free pro-unipotent group $ F_2(k) $ and the group-like elements of $ U\mathfrak{F}_2 $, mapping generators to exponentials and associators.
• Proof via direct computation that the pentagon equation implies the hexagon equations by verifying that the left-hand side of the hexagon equation maps to the corresponding expression under the isomorphism.
• Utilization of the 5-cycle relation in the unipotent completion of the pure braid group $ K_4 $, showing equivalence to the pentagon equation in $ GT $.
• Application of the natural embedding from pro-$l$ and pro-nilpotent completions into $ \prod_l F_2(\mathbb{Q}_l) $ to extend the result to these contexts.
• Definition of $ c_2(f) \in \mathbb{Z}_l $ via the quotient $ F_2^{(l)}(1)/F_2^{(l)}(2) $, enabling the expression $ \lambda = \pm(24c_2(f)+1)^{1/2} $.

Experimental results

Research questions

• RQ1 Does the pentagon equation alone imply the two hexagon equations in the context of Lie algebras and pro-unipotent groups?
• RQ2 Can the parameter $ \mu $ in the hexagon equations be derived from the coefficient $ c_2(\varphi) $ of the associator $ \varphi $, and does this require a field extension?
• RQ3 In the pro-$l$ and pro-nilpotent settings, does the pentagon equation imply the hexagon equations, and what is the role of $ c_2(f) $ in determining the parameter $ \lambda $?
• RQ4 Is the Grothendieck-Teichmüller group $ GT $ defined solely by the pentagon equation, with the hexagon equations being redundant?
• RQ5 Does the pentagon equation in $ GT $ imply the hexagon axioms of braided tensor categories, or is this a weaker condition due to the braid group structure?

Key findings

• The pentagon equation implies the two hexagon equations in the Lie algebra setting, with the parameter $ \mu = \pm(24c_2(\varphi))^{1/2} $, requiring a quadratic extension of the base field.
• In the pro-unipotent setting, the pentagon equation implies the hexagon equations with $ \lambda = \pm(24c_2(f)+1)^{1/2} $, again requiring a quadratic extension.
• The Drinfel'd associator $ \Phi_{KZ} $ satisfies the hexagon equations with $ \mu = \pm 2\pi\sqrt{-1} $, which lies in a quadratic extension of $ \mathbb{R} $, confirming the necessity of such extensions.
• The Grothendieck-Teichmüller group $ GT $ is defined by the pentagon equation alone, with the hexagon equations being consequences, and the group structure is given by $ (\lambda_1,f_1) \circ (\lambda_2,f_2) = (\lambda_1\lambda_2, f) $ with $ f $ defined via conjugation.
• The result extends to pro-$l$ and pro-nilpotent completions, where $ \lambda = \pm(24c_2(f)+1)^{1/2} $ still determines the hexagon equations from the pentagon equation.
• The 5-cycle relation in $ K_4(k) $ is equivalent to the pentagon equation in $ GT $, showing that the pentagon equation captures the essential algebraic structure of $ GT $.