K-Theoretic Hall Algebras

• K-Theoretic Hall algebras are algebraic structures built from the K-theory of moduli stacks of objects like quiver representations and coherent sheaves, with products defined via geometric correspondences.
• They employ techniques such as semi-orthogonal decompositions and PBW-type filtrations to reveal underlying q-deformed symmetric and quantum group structures.
• These algebras bridge geometric methods and representation theory, categorifying extension phenomena and linking to quantum affine and toroidal algebra frameworks.

K-theoretic Hall algebras (KHAs) are algebraic structures encoding the $K$-theory of moduli stacks of objects—such as quiver representations, coherent sheaves on surfaces, or more general geometric data—with multiplication arising from geometric correspondences reflecting their extension theory. In the context of quivers with potential, KHAs extend the cohomological Hall algebra (CoHA) framework of Kontsevich–Soibelman, categorifying representation-theoretic phenomena and quantum group structures, and integrating crucial features such as the PBW theorem, semi-orthogonal decompositions, and explicit relations with quantum affine and toroidal algebras, their modules, and categorifications (Pădurariu, 2019, Pădurariu, 2021, Pădurariu, 2021, Pădurariu, 2021, Varagnolo et al., 2020, Zhu, 4 Nov 2025, Küng et al., 11 Jul 2025).

1. Foundational Definitions and Structure

Given a quiver with potential $(Q,W)$—with $Q=(I,E)$ a finite quiver and $W\in\mathbb{C}Q/[\mathbb{C}Q,\mathbb{C}Q]$—the moduli stack of $d$-dimensional representations is

$$X'(d) = R(d)/G(d), \quad R(d) = \prod_{a\in Q_1} \mathrm{Hom}(\mathbb{C}^{d_{s(a)}},\mathbb{C}^{d_{t(a)}}), \quad G(d) = \prod_{i\in I} GL_{d_i}(\mathbb{C}).$$

The potential $W$ induces a regular function $\mathrm{Tr}(W):X'(d)\rightarrow\mathbb{A}^1$, whose zero fiber $X(d)_0 = \{x\in X'(d) \mid \mathrm{Tr}(W)(x) = 0\}$ is critical. The (homologically $\mathbb{Z}/2\mathbb{Z}$-periodic) category of singularities $\mathsf{D}_{\mathrm{Sg}}(X(d)_0)$, or equivalently the dg-category of matrix factorizations $\mathsf{MF}(X'(d),\mathrm{Tr} W)$, plays the central role: $$K_{\mathrm{crit}}(X(d)) := K_0(\mathsf{D}_{\mathrm{Sg}}(X(d)_0))$$ The $K$-theoretic Hall algebra is the $\mathbb{N}^I$-graded $\mathbb{Q}$-vector space

$$\mathrm{KHA}_{Q,W} = \bigoplus_{d\in\mathbb{N}^I} K_{\mathrm{crit}}(X(d)),$$

admitting an associative, unital convolution product.

2. Hall Multiplication and Convolution Formalism

The associative multiplication arises via a correspondence reflecting short exact sequences: $$X'(d,e) = \{0\to A \to B \,\mid\, \dim A = d,\, \dim B/A = e\}/G(d,e)$$ with

$$q:X'(d,e)\to X'(d)\times X'(e),\quad p:X'(d,e)\to X'(d+e),$$

and the virtual normal bundle $N_p^\vee$. On categories, the product is constructed by a sequence of functorial steps incorporating the Thom–Sebastiani equivalence ($\mathsf{TS}$) and the $K$-theoretic Euler class: $$m_{d,e}(x,y) = p_*\left(q^*(x \boxtimes y) \otimes \Lambda_{-1}(N_p^\vee)\right)$$ This structure is associative by standard base-change arguments and makes $\mathrm{KHA}_{Q,W}$ a braided algebra (Pădurariu, 2019, Pădurariu, 2021, Pădurariu, 2021, Pădurariu, 2021).

3. PBW Filtration and (Deformed) Symmetric Algebra

For symmetric $Q$, a filtration on $\mathrm{KHA}_{Q,W}$ arises from a system of semi-orthogonal decompositions indexed by polytope data, yielding subspaces $M(d)\subset K_{\rm crit}(X(d))$. The PBW theorem, in both its geometric and algebraic incarnations, establishes that the associated graded algebra is a $q$-deformed symmetric (super)algebra: $$\operatorname{gr}^F \mathrm{KHA}_{Q,W} \cong \mathrm{Sym} \Bigl(\bigoplus_{d\in\mathbb{N}^I} M(d)\Bigr)$$ with explicit commutation factors

$$x_{e,v} x_{d,w} = (-1)^{\chi(d,e)} q_f(e,d) x_{d,w} x_{e,v} q_{-g}(e,d),$$

where $\chi(d,e)$ is the Euler form, and the $q_f, q_{-g}$ are explicit kernel factors in $K$-theory (Pădurariu, 2019, Pădurariu, 2021). This structure is extended and refined via semi-orthogonal decompositions using noncommutative resolutions and window subcategories (Pădurariu, 2021, Pădurariu, 2021).

4. Chern Character and Relation to Cohomological Hall Algebras

There is a canonical, functorial Chern character

$$\mathrm{ch}: K_{\mathrm{crit}}(X(d)) \to H_c^\bullet(X(d),\varphi_{\mathrm{Tr} W})$$

which, upon passage to associated gradeds, becomes a morphism of (bi)algebras: $$\mathrm{ch}:\operatorname{gr}^F \mathrm{KHA}_{Q,W} \rightarrow \mathrm{CoHA}_{Q,W}$$ compatible with the perverse and PBW filtrations, mapping $K$-BPS generators to cohomological BPS Lie algebra generators (Pădurariu, 2019, Küng et al., 11 Jul 2025, Lunts et al., 2022). In the symmetric case, this morphism is injective after twist (the Zhang twist), and an equivalence of “locally finite” module categories between KHA and a twisted CoHA can be established (Lunts et al., 2022, Küng et al., 11 Jul 2025).

5. Quantum Group Realizations and Drinfeld Doubles

For framed quivers (i.e., those with added vertices and arrows corresponding to a "framing" vector), the KHA acts naturally on the $K$-theory of Nakajima quiver varieties. In the “tripled” quiver case—where the quiver is augmented to form a 3-Calabi–Yau Ginzburg dg quiver—the KHA is conjecturally (and in key examples, provably) isomorphic to the positive half of the corresponding quantum affine algebra $U_q(\widehat{\mathfrak{g}_Q})$ of Maulik–Okounkov–Smirnov (Pădurariu, 2019, Zhu, 4 Nov 2025, Varagnolo et al., 2020). The Drinfeld double of the KHA produces a Hopf algebra structure closely matching the quantum loop group and, after localizing parameters, leads to isomorphisms with Maulik-Okounkov quantum loop groups (Zhu, 4 Nov 2025, Pădurariu, 2021):

Example	KHA Partner	Quantum Group Counterpart
Jordan quiver, $W=xyz-xzy$	$\mathrm{Ell}^+_{q_1,q_2}$ (Elliptic Hall algebra)	$U_q(\widehat{\mathfrak{gl}_1})^+$
Type $A_n$ tripled quiver	$U_q(\widehat{\mathfrak{sl}}_{n+1})^+$	$U_q(\widehat{\mathfrak{sl}}_{n+1})^+$
General Dynkin tripled	KHA of preprojective algebra	Positive half of quantum toroidal algebra

These identifications are realized via shuffle algebra models, stable envelope operators, and explicit $K$-theory correspondences (Pădurariu, 2021, Varagnolo et al., 2020, Zhu, 4 Nov 2025).

6. Generalizations, Categorification, and New Phenomena

For general (not necessarily tripled or framed) quivers with potential, KHAs exhibit new phenomena:

• Calabi–Yau 3-folds: KHA constructions extend to categories of singularities attached to general CY$_3$ varieties, with the algebra structure reflecting higher-dimensional Donaldson–Thomas invariants (Pădurariu, 2019).
• Categorical BPS / KBPS Algebras: The initial filtration step $M(d)$ produces new "K-theoretic BPS" Lie algebras that interpolate motivic and cohomological BPS structures (Pădurariu, 2019).
• Wall-Crossing and Surface/Curve Theories: Via dimensional reduction, analogous KHAs emerge for local surfaces and curves, capturing wall-crossing and PBW theorems for moduli of Higgs sheaves and local systems (Pădurariu, 2019, Porta et al., 2019, Pădurariu, 2021).
• Torsion Pair Formalism: For abelian and derived categories admitting torsion pairs, one can construct KHAs and their actions on modules associated to the torsion-free part, thereby realizing positive parts of quantum loop algebras and their operators on perverse sheaves or stable pairs (Diaconescu et al., 2022).

7. Techniques: Shuffle Presentations, Filtrations, and Derived Frameworks

A key unifying feature is the realization of KHAs as shuffle-type algebras: symmetric functions in variables indexed by dimension data, equipped with explicit kernel functions derived from the underlying geometry (e.g., K-theory Euler classes, virtual normal bundles) (Zhao, 2019, Pădurariu, 2021, Pădurariu, 2019). Semi-orthogonal decompositions and PBW-type filtrations are constructed categorically, refining to strong algebraic decompositions of KHAs as super-symmetric or deformed symmetric objects generated by primitive components (i.e., intersection $K$-theory classes or matrix factorization contributions) (Pădurariu, 2021, Pădurariu, 2021).

These constructions are deeply tied to categorification strategies, relating Grothendieck groups (KHAs) to the $K$-theory of derived or dg-categories equipped with monoidal structures arising from extension correspondences, forming an interface with categorified Riemann–Hilbert and non-abelian Hodge theory in two-dimensional and higher settings (Porta et al., 2019, Pădurariu, 2021, Pădurariu, 2021).

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References:

• (Pădurariu, 2019) "K-theoretic Hall algebras for quivers with potential"
• (Pădurariu, 2021) "Categorical and K-theoretic Hall algebras for quivers with potential"
• (Pădurariu, 2021) "Generators for K-theoretic Hall algebras of quivers with potential"
• (Pădurariu, 2021) "K-theoretic Hall algebras of quivers with potential as Hopf algebras"
• (Varagnolo et al., 2020) "K-theoretic Hall algebras, quantum groups and super quantum groups"
• (Zhu, 4 Nov 2025) "Maulik-Okounkov quantum loop groups and Drinfeld double of preprojective $K$-theoretic Hall algebras"
• (Küng et al., 11 Jul 2025) "Comparison of Cohomological and K-theoretical Hall algebra"
• (Lunts et al., 2022) "On cohomological and K-theoretical Hall algebras of symmetric quivers"
• (Pădurariu, 2021) "Generators for categorical Hall algebras of surfaces"
• (Diaconescu et al., 2022) "Cohomological Hall algebras and their representations via torsion pairs"
• (Zhao, 2019) "On the $K$-theoretic Hall algebra of a surface"
• (Porta et al., 2019) "Two-dimensional categorified Hall algebras"