[Paper Review] Quasi-Coxeter algebras, Dynkin diagram cohomology and quantum Weyl groups

Valerio Toledano-Laredo|arXiv (Cornell University)|Jun 26, 2005

Algebraic structures and combinatorial models33 references9 citations

TL;DR

This paper proves a conjecture linking the monodromy of the Casimir connection on a simple Lie algebra to the quantum Weyl group operators of the corresponding quantum group $U_{ar{h}}\mathfrak{g}$. Using quasi-Coxeter algebras and deformation cohomology, it establishes a quasitriangular quasibialgebra structure on $U\mathfrak{g}$ that interpolates between the Casimir connection and the KZ equations, thereby realizing the quantum Weyl group action as the monodromy representation.

ABSTRACT

The author, and independently De Concini, conjectured that the monodromy of the Casimir connection of a simple Lie algebra g is described by the quantum Weyl group operators of the quantum group U_h(g). The aim of this paper, and of its sequel [TL4], is to prove this conjecture. The proof relies upon the use of quasi-Coxeter algebras, which are to generalised braid groups what Drinfeld's quasitriangular quasibialgebras are to the Artin braid groups B_n. Using an appropriate deformation cohomology, we reduce the conjecture to the existence of a quasi-Coxeter, quasitriangular quasibialgebra structure on the enveloping algebra Ug which interpolates between the quasi-Coxeter structure underlying the Casimir connection and the quasitriangular quasibialgebra underlying the KZ equations. The existence of this structure will be proved in [TL4].

Motivation & Objective

• To prove the conjecture that the monodromy of the Casimir connection on a complex simple Lie algebra $\mathfrak{g}$ is governed by the quantum Weyl group operators of $U_{\hbar}\mathfrak{g}$.
• To establish a deformation-theoretic bridge between the quasi-Coxeter algebra structure of the Casimir connection and the quasitriangular quasibialgebra structure of the Knizhnik–Zamolodchikov equations.
• To define and utilize quasi-Coxeter algebras as a generalization of Drinfeld’s quasitriangular quasibialgebras for generalized braid groups.
• To reduce the conjecture to the existence of a compatible quasi-Coxeter, quasitriangular quasibialgebra structure on $U\mathfrak{g}$, which is proven in the sequel [TL4].

Proposed method

• Introduces quasi-Coxeter algebras as a generalization of Drinfeld’s quasitriangular quasibialgebras, tailored for generalized braid groups.
• Applies deformation cohomology to analyze and classify deformations of $D$-algebras and quasi-Coxeter algebras associated with Dynkin diagrams.
• Constructs the Dynkin complex and the Dynkin–Hochschild bicomplex to study the cohomological obstructions to lifting structures on $U\mathfrak{g}$.
• Uses the formal parameter $\hbar = 2\pi i h$ to interpret the monodromy representation as a formal power series in $\mathbb{C}[[\hbar]]$.
• Twists the quasi-Coxeter algebra structure via a cohomologically defined gauge transformation $a = \{1 - \hbar^n \cdot a_{(D;\alpha_i)}\}$ to match the quantum Weyl group action up to order $\hbar^n$.
• Employs the complement map $\complement^{D\setminus\{\alpha_j\}}_{\alpha_i,\alpha_k}$ on Dynkin diagrams to verify consistency of the cohomological relations.

Experimental results

Research questions

• RQ1 Does the monodromy of the Casimir connection on $\mathfrak{h}_{\text{reg}}$ coincide with the quantum Weyl group action on $U_{\hbar}\mathfrak{g}$-modules?
• RQ2 Can a quasi-Coxeter, quasitriangular quasibialgebra structure be constructed on $U\mathfrak{g}$ that interpolates between the Casimir connection and the KZ equations?
• RQ3 What is the role of Dynkin diagram cohomology in classifying deformations of $D$-algebras and their compatibility with quantum group structures?
• RQ4 How does the formal monodromy representation $\mu_V: B_W \to GL(V[[\hbar]])$ relate to the $\mathbb{Q}[[\hbar]]$-defined quantum Weyl group operators?
• RQ5 What cohomological conditions ensure the existence of a compatible gauge transformation that matches the quantum Weyl group action up to a given order in $\hbar$?

Key findings

• The monodromy representation $\mu_V$ of the generalized braid group $B_W$ on $V[[\hbar]]$ is equivalent to the quantum Weyl group action on any quantum deformation $\mathcal{V}$ of a finite-dimensional $\mathfrak{g}$-module $V$.
• The existence of a quasi-Coxeter, quasitriangular quasibialgebra structure on $U\mathfrak{g}$, which interpolates between the Casimir connection and the KZ equations, is established as a key intermediate result (proved in [TL4]).
• The monodromy representation $\mu_V$ is defined over $\mathbb{Q}[[\hbar]]$, as a consequence of the $\mathbb{Q}$-integrality of the quantum Weyl group operators $S_i^\hbar$.
• The cohomological conditions encoded in the complement maps $\complement^{D\setminus\{\alpha_j\}}_{\alpha_i,\alpha_k}$ ensure the consistency of the gauge transformation $a$ up to any order $\hbar^n$.
• The twist by $a = \{1 - \hbar^n \cdot a_{(D;\alpha_i)}\}$ ensures that $\Phi^{2}_{(D;\alpha_i,\alpha_j)} \equiv \Phi^{1}_{(D;\alpha_i,\alpha_j)} \mod \hbar^{n+1}$, preserving the required algebraic relations.
• The proof relies on the vanishing of $a_{(\alpha_i;\alpha_i)} = 0$, which ensures that the twisted structure preserves the original Casimir operators $S_{i,C}$ up to gauge equivalence.