Derived Hall Algebras Overview

• Derived Hall algebras are algebraic structures attached to triangulated categories that encode extension data and extend classical Hall algebras.
• They define multiplication through exact triangles and precise structure constants, bridging homological methods with quantum algebra.
• These algebras connect representation theory, moduli spaces, and quantum invariants, providing frameworks for categorification and explicit algebraic realizations.

A derived Hall algebra is an algebraic structure attached to a finitary triangulated or derived category, generalizing the classical Ringel–Hall algebra of an abelian category by encoding the extension structure of its triangulated category. Derived Hall algebras provide a framework that realizes and categorifies quantum groups, their doubles, and related objects, and have deep connections to representation theory, categorification, moduli theory, and quantum invariants.

1. Foundations: Definition and Axiomatic Structure

Let $k$ be a finite field, and let $\mathcal{T}$ be a $k$-linear, Hom-finite, Krull–Schmidt, triangulated category, e.g., the bounded derived category $D^b(A)$ of a finite-dimensional $k$-algebra $A$ or the derived category of coherent sheaves on a smooth variety. The derived Hall algebra $H(\mathcal{T})$ is formulated as follows:

• The underlying vector space has basis $\{[X] \mid X \in \mathrm{Iso}(\mathcal{T})\}$.
• The product is defined by

$[M]\cdot[N]=\sum_{[L]} F_{M,N}^L\,[L]$

with structure constants

$$F_{M,N}^L = \frac{|\mathrm{Hom}(M,L)_{\mathrm{naive}}|\ {L,N}} {|\mathrm{Aut}(M)| \, \{M,M\} \, |\mathrm{Aut}(N)|}$$

where $|\mathrm{Hom}(M,L)_{\mathrm{naive}}|$ counts morphisms $f \colon M \to L$ such that $\mathrm{cone}(f) \cong N$, and

$$\{X,Y\} = \prod_{n\geq 0} |\mathrm{Hom}(X,\Sigma^n Y)|^{(-1)^n}.$$

Associativity of this product follows from Toën and equivalently from Green's four-term extension formula (Lin, 2024).

In the case where $\mathcal{T}=D^b(\mathcal{A})$ for a hereditary abelian category $\mathcal{A}$, $F_{M,N}^L$ recovers the classical Hall algebra structure constants in degree zero and the derived extension structure for objects with nontrivial shifts (Xiao et al., 2012, Lin, 2024).

2. Structure Theory: Generators, Relations, and Presentations

The presentation of the derived Hall algebra depends intricately on the structure of the underlying triangulated category.

• For hereditary categories and their bounded derived categories, the derived Hall algebra admits a presentation with generators labeled by isomorphism classes of objects (e.g., indecomposable complexes and their shifts) and relations governed by Serre-type and commutation relations reflecting the configuration of exact triangles (Hernandez et al., 2011).
• In the case of one-cycle gentle algebras of infinite global dimension, the derived Hall algebra $DH(A)$ is presented by explicit families of generators $x_{k,i}$ (for each vertex and integer grading) and $z_{k,i}$ (central cycle elements), with quadratic and cubic relations generalizing quantum Serre relations but with an infinite, nontruncating structure (Bobinski et al., 2019).

For root categories and periodic orbit categories, the derived Hall algebra incorporates two copies of the classical Hall algebra—shifted by $[1]$—and admits a realization as the Drinfeld double of the classical (abelian) Hall algebra (Zhang, 2022, Chen et al., 2023).

Table: Key Families of Derived Hall Algebra Presentations

Category type	Generators	Relations
$D^b(\mathrm{mod}\,kQ)$ (Dynkin quiver)	$z_{i,m}= [S_i[m]]$	Quantum Serre, commutation, adjacent-degree (Hernandez–Leclerc) relations (Hernandez et al., 2011)
One-cycle gentle, infinite gldim	$x_{k,i}$, $z_{k,i}$	Quadratic/cubic, inter-string, and cycle relations (Bobinski et al., 2019)
$D^b(\mathcal{A})/[m]$ ($m$-periodic)	$u_{M[i]}$, $K_{\alpha,i}$	Derived-Peng–Riedtmann formulas, torus relations (Zhang, 2023)
Perfect derived Fukaya category	$E_{i,n}$ (arcs, gradings)	Skein-type, convolution, far-commutation (Cooper et al., 2017)

3. Periodic and Extended Variants

The derived Hall algebra has rich periodic and extended (quantum-torus) generalizations:

• $m$-periodic derived Hall algebras: For an odd $m$, one defines the Hall algebra of the $m$-periodic derived category $D_m(\mathcal{A})$ with a multiplication governed by generalized Hall numbers indexed by $m$-periodic complexes. The extended version, including quantum torus elements $K_{\alpha,i}$, provides a unified structure containing both the derived Hall and semi-derived Hall algebras, crucial for the categorification of quantum groups (Zhang, 2023, Zhang et al., 2023).
• Delta-Hall algebras and $\imath$-quantum groups: The $\Delta$-Hall algebra of a hereditary abelian category provides a multiplication encoding triple extension data (three-cycles) and is isomorphic to the 1-periodic derived Hall algebra. With additional central and torus extensions, this recovers the semi-derived and $\imath$-quantum groups (Chen et al., 2022).

4. Drinfeld Doubles, Quantum Groups, and Connections

Derived Hall algebras categorify quantum group structures and realize Drinfeld doubles:

• Drinfeld double realization: For abelian categories $\mathcal{A}$, their root category derived Hall algebras are isomorphic to the Drinfeld double of the classical Hall algebra $D(H(\mathcal{A}))$, allowing for the explicit modeling of both positive and negative parts of a quantum group in a categorical setting (Zhang, 2022, Chen et al., 2023, Xu et al., 2019).
• Quantum group isomorphisms: For hereditary types, presentations in terms of quantum Serre relations show that $DH(A)$ (in the finite global dimension case) realizes $U^+_q(\widehat{\mathfrak{g}})$ for the underlying affine type (Bobinski et al., 2019, Hernandez et al., 2011).
• Categorification and canonical bases: Natural bases of derived Hall algebras are matched to (dual) canonical bases of quantum enveloping algebras via Lusztig's geometric construction or quiver variety techniques (Hernandez et al., 2011).

5. Semi-derived, Motivic, and Geometric Hall Algebras

Generalizations and extensions involve further categorical and motivic enhancements:

• Semi-derived Hall algebras: For Frobenius categories, the semi-derived Hall algebra is the localization of their Hall algebra at all projective-injectives. When the stable category is triangulated and Hom-finite, the semi-derived Hall algebra is isomorphic, up to a twisted group algebra, to the derived Hall algebra of the stable (triangulated) category (Gorsky, 2014).
• Motivic derived Hall algebras: For ind-constructible (dg- or $A_\infty$-) Calabi–Yau categories, motivic Hall and motivic derived Hall algebras are constructed from stack function spaces, and are isomorphic via the motivic Riedtmann–Peng formula. This enables wall-crossing and the formulation of Donaldson–Thomas invariants categorically (Xiao et al., 2012, Toda, 2016).
• Applications to Fukaya and perverse sheaf categories: For (partially wrapped) Fukaya categories of surfaces, the derived Hall algebra encoding the morphism and extension data of graded arcs gives rise to skein-type algebraic structures, enabling a recursive computation matching HOMFLY-PT skein relations and mapping to quantum group presentations (Cooper et al., 2017).

6. Structural Theorems and Representative Examples

Key theorems include:

• Associativity via Green's formula: Associativity of the derived Hall product is equivalent to the validity of Green’s (four-term extension) formula in the underlying abelian Hall algebra, and is established by reduction to the octahedral axiom and categorical counting of triangles (Lin, 2024, Xiao et al., 2012).
• Tensor decomposition: Extended (or modified, or semi-derived) Hall algebras decompose as a tensor product of the derived Hall algebra and a quantum torus generated by $K$-theoretic data, facilitating explicit comparison and invariance under derived equivalence (Lin et al., 2017, Gorsky, 2014).
• Explicit generators and relations: See the complete presentations for one-cycle gentle algebras (Bobinski et al., 2019), surfaces (Cooper et al., 2017), and root categories (Zhang, 2022, Chen et al., 2023).

Table: Derived Hall Algebra—Key Comparison Points

Feature	Classical Hall Algebra	Derived Hall Algebra
Category	Abelian (finitary, hereditary, etc.)	Triangulated, dg-category, periodic
Generators	Isoclasses of objects in $\mathcal{A}$	Isoclasses in $D^b(\mathcal{A})$
Product	Extension short exact sequences	Counting exact triangles (cones)
Quantum group content	$U_q^+(\mathfrak{g})$	Drinfeld double $U_q(\mathfrak{g})$

7. Outlook: Open Directions and Categorical Unification

Contemporary developments focus on extending the theory to:

• Even-periodic and non-hereditary situations: Construction of derived Hall algebra analogues for $D^b(\mathcal{A})/[m]$ with $m$ even, and for categories with more general homological properties, is active and open (Zhang, 2023, Zhang et al., 2023).
• Motivic and categorified invariants: Motivic Hall algebras and their integration maps categorize wall-crossing phenomena and Donaldson–Thomas type invariants in derived, stable, or Fukaya categorical settings (Xiao et al., 2012, Toda, 2016).
• Relations to cohomological Hall algebras and shifted symplectic geometry: The extension of derived Hall algebra techniques to categorifications in the sense of cohomological Hall algebras (CoHAs), especially for 3-Calabi–Yau categories and moduli of (semi-)Schur objects, is an area of ongoing research (Toda, 2016).

The derived Hall algebra thus occupies a central position at the interface of representation theory, homological algebra, categorification, and quantum algebra, offering a foundational categorical model for quantum groups, their doubles, and higher representation-theoretic and geometric structures (Hernandez et al., 2011, Xiao et al., 2012, Bobinski et al., 2019, Chen et al., 2023).