[Paper Review] A note on Bridgeland's Hall algebra of two-periodic complexes

Shintarou Yanagida|arXiv (Cornell University)|Jul 4, 2012

Algebraic structures and combinatorial models9 references4 citations

TL;DR

This paper establishes that Bridgeland's Hall algebra of two-periodic complexes over a hereditary abelian category with finite global dimension is isomorphic to the Drinfeld double of the ordinary extended Hall bialgebra of the category. The isomorphism arises through a twisting by the Euler form and localization at acyclic complexes, providing a structural link between categorified Hall algebras and quantum group constructions.

ABSTRACT

We show that the Hall algebra of two-periodic complexes, which is recently introduced by T.Bridgeland, coincides with the Drinfeld double of the ordinary Hall bialgebra.

Motivation & Objective

• To establish a precise algebraic isomorphism between Bridgeland's Hall algebra of two-periodic complexes and a known quantum group construction.
• To clarify the algebraic structure of the Hall algebra associated with $Δ$-graded complexes in abelian categories over finite fields.
• To demonstrate that the construction is functorial under derived equivalences, ensuring invariance under derived categories.
• To provide a conceptual framework linking categorified Hall algebras and Drinfeld doubles in the context of derived categories.
• To extend the applicability of Hall algebra techniques to derived categories via the Drinfeld double construction.

Proposed method

• Define the Hall algebra of $Δ$-graded complexes ($Δ = \mathbb{Z}/2\mathbb{Z}$) in an abelian category $Α$ over a finite field.
• Construct the twisted Hall algebra $\cal{H}_{\mathrm{tw}}(\cal{C}(\cal{P}))$ using the Euler form of $\cal{A}$.
• Localize this twisted algebra at acyclic complexes to define Bridgeland's algebra $\cal{DH}(\cal{A})$.
• Establish an isomorphism between $\cal{DH}(\cal{A})$ and the Drinfeld double of the extended Hall bialgebra $\widetilde{\cal{H}}(\cal{A})$.
• Use the theory of Drinfeld doubles and the structure of hereditary categories to verify the isomorphism at the level of algebraic relations and generators.
• Apply results from Ringel and Cramer to ensure functoriality and invariance under derived equivalences.

Experimental results

Research questions

• RQ1 Is Bridgeland's Hall algebra of two-periodic complexes isomorphic to a known quantum group construction such as the Drinfeld double?
• RQ2 How does the Drinfeld double structure emerge from the Hall algebra of complexes in a hereditary category?
• RQ3 What is the role of the Euler form and localization at acyclic complexes in realizing the Drinfeld double?
• RQ4 Can the construction be extended to be invariant under derived equivalences of the underlying category?
• RQ5 What is the action of the group $\operatorname{SL}_2(\mathbb{Z})$ on the Hall algebra in the case of elliptic curves?

Key findings

• The Hall algebra $\cal{DH}(\cal{A})$ of two-periodic complexes is isomorphic to the Drinfeld double of the extended Hall bialgebra $\widetilde{\cal{H}}(\cal{A})$ as an associative algebra.
• The isomorphism holds under the conditions: $\cal{A}$ is hereditary, of finite global dimension, has enough projectives, and nonzero objects define nonzero classes in $K(\cal{A})$.
• The construction is functorial: derived equivalences $\Phi: D^b(\cal{A}) \xrightarrow{\sim} D^b(\cal{B})$ induce algebra isomorphisms $\Phi^{\cal{DH}}: \cal{DH}(\cal{A}) \xrightarrow{\sim} \cal{DH}(\cal{B})$, compatible with composition.
• For $\cal{A} = \operatorname{Coh}(C)$ on an elliptic curve $C$, the group $\operatorname{Auteq}(D^b(\cal{A}))$ maps to $\operatorname{Aut}^{\cal{DH}}(\cal{T})$, with $\operatorname{SL}_2(\mathbb{Z})$ acting as automorphisms on $\cal{DH}(\cal{T})$.
• The action of $\operatorname{SL}_2(\mathbb{Z})$ on $\cal{DH}(\cal{T})$ is isomorphic to the $\operatorname{SL}_2(\mathbb{Z})$-automorphisms found in other works on AGT correspondence.
• The kernel of the map $DH: \operatorname{Auteq}(\cal{T}) \to \operatorname{Aut}^{\cal{DH}}(\cal{T})$ is $\mathbb{Z}/2\mathbb{Z}$, corresponding to the involution $*$ on $\cal{DH}(\cal{T})$.