[Paper Review] Categorical construction of A,D,E root systems

Alexander Kirillov, Jaimal Thind|arXiv (Cornell University)|Jul 15, 2010

Algebraic structures and combinatorial models24 references3 citations

TL;DR

This paper presents a canonical, orientation-independent categorical construction of simply-laced A, D, E root systems using a translation quiver $\tilde{\Gamma}$ and a 2-periodic triangulated category $\mathcal{C} = D/\langle T^2 \rangle$. The roots are realized as isomorphism classes of indecomposable objects in $\mathcal{C}$, with the inner product and root lattice defined via the Euler form. The key result is a periodicity property of the preprojective algebra, leading to a canonical Coxeter element given by translation by $2h$, where $h$ is the Coxeter number.

ABSTRACT

Let Γbe a Dynkin diagram of type A,D,E and let R denote the corresponding root system. In this paper we give a categorical construction of R from Γ. Instead of choosing an orientation of Γand studying representations of the associated quiver, we study representations of a canonical quiver \Gammahat associated to Γ. This construction is very closely related to the preprojective algebra of Γ. In particular, the construction gives a certain periodicity result about the preprojective algebra.

Motivation & Objective

• To provide a canonical, orientation-independent construction of A, D, E root systems using category theory.
• To eliminate the dependence on arbitrary choices of quiver orientation or simple roots in standard root system constructions.
• To establish a categorical framework where roots correspond to indecomposable objects in a triangulated category with explicit inner product and root lattice structure.
• To prove a periodicity result for the Koszul complex of the preprojective algebra in the Dynkin case, linking it to the Coxeter number.

Proposed method

• Construct a translation quiver $\tilde{\Gamma} \subset \Gamma \times \mathbb{Z}$ from the Dynkin diagram $\Gamma$, and define a triangulated category $D$ as a full subcategory of the derived category $D(\tilde{\Gamma})$.
• Use the Auslander-Reiten quiver of $D$ to identify indecomposable objects with vertices in $\tilde{\Gamma}^{\text{op}}$, and relate them to the mesh category via the preprojective algebra.
• Define the quotient category $\mathcal{C} = D / T^2$, where $T$ is the translation functor, to obtain a 2-periodic triangulated category.
• Establish equivalences between $\mathcal{C}$ and derived categories $D^b(\text{Rep}(\Gamma, \Omega_h))/T^2$ for any height function $h$, ensuring compatibility with BGP reflection functors.
• Use the graphical description of the Koszul complex of the preprojective algebra, visualizing elements as paths with 'jumps' in $\tilde{\Gamma}$, to compute $\text{RHom}$ complexes.
• Prove that the category $\mathcal{C}$ admits a canonical Coxeter element $C$ given by $X \mapsto X(-2)$, and that $C^{2h} = \text{id}$, reflecting the Coxeter number $h$.

Experimental results

Research questions

• RQ1 Can a canonical, orientation-independent construction of A, D, E root systems be achieved via category theory?
• RQ2 How does the preprojective algebra’s Koszul complex exhibit periodicity in the Dynkin case?
• RQ3 What is the role of the translation quiver $\tilde{\Gamma}$ and its derived category in realizing roots as indecomposable objects?
• RQ4 How is the Coxeter element realized categorically in the 2-periodic quotient category $\mathcal{C} = D/T^2$?
• RQ5 Can the Euler form and inner product on the Grothendieck group be explicitly described in terms of path algebras on $\tilde{\Gamma}^{\text{cyc}}$?

Key findings

• The category $\mathcal{C} = D/T^2$ is 2-periodic, satisfying $T^2 = \text{id}$, and satisfies $F(2h) \cong F$ for all objects $F$, with $h$ the Coxeter number.
• The Grothendieck group $K$ of $\mathcal{C}$ is isomorphic to the root lattice, and the set of indecomposable classes $\text{Ind} \subset K$ corresponds bijectively to the roots of the A, D, E root system.
• The Coxeter element $C$ is realized as the functor $X \mapsto X(-2)$, and acts on the Auslander-Reiten quiver $\tilde{\Gamma}^{\text{cyc}}$ as translation $\tau: (i,n) \mapsto (i,n+2)$.
• The inner product on $K$ is given by $(X,Y) = \langle X,Y \rangle_C + \langle Y,X \rangle_C$, where $\langle X,Y \rangle_C = \dim \text{RHom}(X,Y)$ is the Euler form.
• The category $\mathcal{C}$ satisfies Serre duality: $\text{Hom}(X,Y) \cong (\text{Ext}^1(Y,X(-2)))^*$, and $\text{Hom}(X_q,X_{q'}) \cong \text{Path}(q',q)/J$, with $J$ an explicitly described ideal.
• A periodicity result holds: $A_{i,j;l}^{\bullet+2} \simeq A_{i,j;l+2h}^{\bullet}$, implying $H^k(A_{i,j;l}) \cong H^{k+2}(A_{i,j;l+2h})$, which reflects the $2h$-periodicity of the preprojective algebra in the Dynkin case.