Hall algebras of even-periodic triangulated categories and the structures left open by the odd-periodic construction: the 2-periodic/root-category problem and the Z2-graded structure of the 3-periodic Hall algebra

The Hall algebra construction of Xu and Chen equips an odd-periodic triangulated category (period t > 1 odd, under finiteness conditions) with an associative algebra whose structure constants count triangles weighted by alternating homological Euler factors, generalizing Toën's derived Hall algebra and the Xiao–Xu construction. The construction provably breaks down for even period — most importantly the 2-periodic root categories at the heart of the Peng–Xiao and Bridgeland programs — because its identities and normalizations are defined only for odd t (the source assumes an odd period throughout) and because such categories violate the homological finiteness assumptions. The open directions recorded in that paper are: defining Hall algebras for even-periodic (especially 2-periodic) triangulated categories, and determining the Z2-graded Lie algebra structure arising from the 3-periodic Hall algebra when the Grothendieck group is Z2-graded (Remark 3.5 of the source). Recent work (2022–2025) resolved the headline 2-periodic problem for m-periodic derived categories of finitary hereditary abelian categories, via both intrinsic triangle-counting and extended-basis constructions, and a third, homotopy-cardinality route (Gorsky–Haiden, 2024) covers odd Calabi–Yau categories, including root categories of 1-Calabi–Yau abelian categories; the general theory of the even-periodic, even-Calabi–Yau and cluster-type/orbit categories not covered by these constructions and the Z2-graded structure of the 3-periodic Hall algebra remain open.

Background

A Hall algebra is an associative algebra built from an abelian category over a finite field: it has a basis indexed by isomorphism classes of objects, and its structure constants count short exact sequences, i.e. the number of ways one object sits between two others. Ringel and Green showed that for a hereditary algebra the (twisted) Hall algebra realizes the positive part of a quantized enveloping algebra — a one-parameter deformation of the universal enveloping algebra of a semisimple or Kac–Moody Lie algebra, the latter being the Lie algebras attached to generalized Cartan matrices that generalize the finite-dimensional semisimple ones — which made Hall algebras a principal tool for realizing quantum groups and Kac–Moody algebras categorically. For triangulated categories the naive counting of triangles does not directly produce an associative algebra; Toën (Duke Math. J. 135, 2006) defined the derived Hall algebra of a differential graded category (a category whose morphism sets form chain complexes of vector spaces) satisfying strong homological finiteness conditions (essentially: Ext groups vanish in almost all degrees so that an alternating Euler-type product over shifts is finite), and Xiao–Xu (Duke Math. J. 143, 2008) extended the construction to triangulated categories satisfying analogous finiteness conditions, with structure constants obtained from triangle counts normalized by the alternating product of Hom-group cardinalities over all shifts. A triangulated category with shift functor $\Sigma$ is called t-periodic if $\Sigma^t$ is isomorphic to the identity. Xu and Chen (Algebr. Represent. Theory 16, 2013) showed that for odd t > 1, under mild finiteness conditions (finite-dimensional Hom spaces, local endomorphism rings of indecomposables), the alternating product over the t shifts is automatically finite and the identities needed for associativity hold, so every odd-periodic triangulated category carries a Hall algebra; they computed the 3-periodic case explicitly. The oddness is essential: the key identity pairs each shift $i$ with the shift $t-i$, and the pairing of the alternating signs is compatible only for odd t — the source assumes throughout that t is odd, the square-root Euler factors normalizing its structure constants (Lemma 2.2, Theorem 2.5) are well-defined exactly in the odd case, and later literature confirms that the proof approach cannot be applied to even periods (Zhang, J. Pure Appl. Algebra 2025). Even-periodic categories, above all the 2-periodic ones, fail the homological finiteness assumptions outright, so neither Toën's nor Xiao–Xu's method applies. The 2-periodic case is the important one: the root category of a hereditary abelian category $\mathcal{A}$ (the bounded derived category modulo $\Sigma^2$) is 2-periodic, and Peng–Xiao (Invent. Math. 140, 2000) constructed simple Lie algebras from root categories at the Lie level, while Bridgeland (Ann. Math. 177, 2013) realized whole quantized enveloping algebras via a localized Hall algebra of 2-periodic complexes — but no Hall algebra of the root category as a triangulated category itself was defined. The structural target of these comparison theorems is typically the Drinfeld double, the canonical braided Hopf algebra combining a Hopf algebra with its vector-space dual, which for Hall algebras reconstructs the full quantized enveloping algebra from its positive half. Keller–Yang–Zhou (JLMS 80, 2009) computed the Hall algebra of the algebraic triangulated category $\mathcal{S}_w$ generated by a w-spherical object (an object whose Ext groups vanish in all degrees except 0 and w, generalizing a sphere in topology); its orbit categories — categories obtained by identifying each object with its image under a fixed autoequivalence, here a power of the shift — include 2-periodic categories and cluster-type/orbit categories (orbit categories under a cluster-like autoequivalence, such as cluster categories and cluster tubes — fundamental examples of Calabi–Yau categories, i.e. triangulated categories whose Serre duality functor — the categorical analogue of Poincaré duality — is a power of the shift). Subsequent work settled important special cases of the even-periodic problem: Zhang and Chen–Lu–Ruan (Israel J. Math. 2025) constructed associative Hall algebras for root categories of finitary hereditary abelian categories, each isomorphic to the Drinfeld double of the Hall algebra of $\mathcal{A}$; Zhang (J. Pure Appl. Algebra 2025) extended this to the m-periodic derived categories $D_m(\mathcal{A})$ for every $m \geq 1$, even $m$ included; Gorsky (IMRN 2018) had earlier constructed semi-derived Hall algebras for stable categories of Frobenius categories with enough projectives, covering the corresponding 2-periodic derived categories; and Gorsky–Haiden (2024) defined intrinsic Hall algebras for odd Calabi–Yau dg-categories via homotopy cardinality, a route that includes root categories of 1-Calabi–Yau abelian categories. The general even-Calabi–Yau and general even-periodic categories are not covered by any of these constructions. Against this backdrop, the open directions recorded in the source paper survive as follows: extend the construction to the remaining even-periodic categories; and deal with the Z2-graded Lie algebras arising from the 3-periodic Hall algebra $\mathcal{H}(D_3(\mathcal{A}))$, whose Grothendieck group (the free abelian group on isomorphism classes of objects modulo the relations imposed by exact triangles; the natural grading group of a Hall algebra) is $\mathbb{Z}_2$-graded — a direction the source records as of interest (Remark 3.5). Beyond these two source directions, the same construction is open for the adjacent even-Calabi–Yau and cluster-type/orbit categories. As a related open strand not among the two directions the source records, defining an associative Hall algebra for the cluster category of a spherical object and relating it to the Keller–Yang–Zhou algebra (JLMS 80, 2009) remains open; the known result there is only the Lie-level one of Fu–Yang (JLMS 85, 2012).

Problem Statement

Complete the Hall algebra theory of periodic triangulated categories left open by the odd-periodic construction of Xu–Chen. The task pursues the two directions the source records — the even-periodic construction and the Z2-graded structure — and notes the adjacent even-Calabi–Yau and cluster-type/orbit families as an extension of the first. (1) Construction for the three target families, the core: for the three families of triangulated categories that do not yet carry an associative Hall algebra, namely the even-periodic categories (those with $\Sigma^t \cong \mathrm{id}$ for even $t$, e.g. the 2-periodic orbit categories $\mathcal{S}_w/\Sigma^2$), the even-Calabi–Yau categories (those whose Serre functor is an even power of the shift) and the cluster-type/orbit categories (orbit categories of a derived category under a cluster-like autoequivalence — typically the shifted inverse Auslander–Reiten translation τ^{-1}[1] or its tube analogue, not a shift power, so the shift-orbit categories $D_m(\mathcal{A})$ fall under the cited exclusions rather than this family — e.g. cluster categories of acyclic quivers and of w-spherical objects, and cluster tubes), each taken with its defining structural handle (the shift functor, the Serre functor, or the orbit structure) and under the finiteness conditions that make the structure constants finite (finite-dimensional Hom spaces, local endomorphism rings of indecomposables, and finiteness of the alternating Hom-cardinality products; these conditions define the family, and the unrestricted class of all triangulated categories is not in scope), the cases already carrying an associative Hall algebra by a cited construction being excluded (the odd-periodic categories by Xu–Chen; the root and m-periodic derived categories $D_m(\mathcal{A})$ of finitary hereditary abelian categories by Zhang and Chen–Lu–Ruan; the stable categories of Frobenius categories with enough projectives, covered by the semi-derived and derived Hall algebras of Gorsky; and the odd-Calabi–Yau categories covered by the homotopy-cardinality construction of Gorsky–Haiden): (a) define an associative Hall algebra with finite structure constants, determining those constants; (b) prove well-definedness and associativity, resolving the obstruction that confines the source's construction to odd periods; and (c) determine the isomorphism type of the resulting algebra and its relation to known algebras (Drinfeld doubles, quantized enveloping algebras, or otherwise). The three families above are the full target class, the listed items being illustrative instances rather than an exhaustive list. The even-periodic family is the target in general, under the stated finiteness conditions: an individual instance is a special case of the family rather than the family itself, and a construction that needs finiteness assumptions beyond those stated must establish them for the even-periodic family in general. The three sub-goals (a)–(c) apply to the even-periodic family; the even-Calabi–Yau and cluster-type/orbit families are an explicit extension of the source's first direction. Directions (1) and (2) are posed jointly. (2) Z2-graded structure: determine the Z2-graded Lie algebra structure of the 3-periodic Hall algebra $\mathcal{H}(D_3(\mathcal{A}))$ — its graded pieces and commutator bracket, and its relation to known Lie algebras — induced by the $\mathbb{Z}_2$-grading of the Grothendieck group (since X = X[3] in $D_3(\mathcal{A})$), the direction the source records as of interest (Remark 3.5 of arXiv:1001.4353). These deliverables jointly serve one objective — extending the periodic Hall algebra framework and its quantum-group realizations from odd periods to the remaining even-periodic settings, with the adjacent even-Calabi–Yau and cluster-type/orbit settings as an extension — with deliverable (2) a structural development of the same framework that the source records alongside the construction problem.

References

1. Fan Xu and Xueqing Chen, Hall Algebras of Odd Periodic Triangulated Categories, Algebras and Representation Theory 16 (2013), no. 3, 673–687, doi:10.1007/s10468-011-9325-z, https://doi.org/10.1007/s10468-011-9325-z (preprint: arXiv:1001.4353, https://arxiv.org/abs/1001.4353)
2. Jie Xiao and Fan Xu, Hall algebras associated to triangulated categories, Duke Mathematical Journal 143 (2008), no. 2, 357–373, doi:10.1215/00127094-2008-021, https://doi.org/10.1215/00127094-2008-021
3. B. Toën, Derived Hall algebras, Duke Mathematical Journal 135 (2006), no. 3, 587–615, doi:10.1215/s0012-7094-06-13536-6, https://doi.org/10.1215/s0012-7094-06-13536-6 (preprint: arXiv:math/0501343, https://arxiv.org/abs/math/0501343)
4. Liangang Peng and Jie Xiao, Triangulated categories and Kac–Moody algebras, Inventiones Mathematicae 140 (2000), no. 3, 563–603, doi:10.1007/s002220000062, https://doi.org/10.1007/s002220000062
5. Bernhard Keller, Dong Yang and Guodong Zhou, The Hall algebra of a spherical object, Journal of the London Mathematical Society 80 (2009), no. 3, 771–784, doi:10.1112/jlms/jdp054, https://doi.org/10.1112/jlms/jdp054
6. Tom Bridgeland, Quantum groups via Hall algebras of complexes, Annals of Mathematics 177 (2013), no. 2, 739–759, doi:10.4007/annals.2013.177.2.9, https://doi.org/10.4007/annals.2013.177.2.9
7. Mikhail Gorsky, Semi-derived and derived Hall algebras for stable categories, International Mathematics Research Notices 2018 (2018), 138–159, doi:10.1093/imrn/rnv325, https://doi.org/10.1093/imrn/rnv325 (preprint: arXiv:1409.6798, https://arxiv.org/abs/1409.6798)
8. Mikhail Gorsky and Fabian Haiden, Counting in Calabi–Yau categories, with applications to Hall algebras and knot polynomials, arXiv:2409.10154, https://arxiv.org/abs/2409.10154
9. Changjian Fu and Dong Yang, The Ringel–Hall Lie algebra of a spherical object, Journal of the London Mathematical Society 85 (2012), no. 2, 511–533, doi:10.1112/jlms/jdr064, https://doi.org/10.1112/jlms/jdr064 (preprint: arXiv:1103.1241, https://arxiv.org/abs/1103.1241)
10. Haicheng Zhang, Hall algebras associated to root categories, Israel Journal of Mathematics 270 (2025), no. 1, 413–448, doi:10.1007/s11856-025-2825-z, https://doi.org/10.1007/s11856-025-2825-z (preprint: arXiv:2210.12351, https://arxiv.org/abs/2210.12351)
11. Jiayi Chen, Ming Lu and Shiquan Ruan, Derived Hall algebras of root categories, Israel Journal of Mathematics 270 (2025), no. 1, 185–228, doi:10.1007/s11856-025-2770-x, https://doi.org/10.1007/s11856-025-2770-x (preprint: arXiv:2303.01670, https://arxiv.org/abs/2303.01670)
12. Haicheng Zhang, Periodic derived Hall algebras of hereditary abelian categories, Journal of Pure and Applied Algebra 229 (2025), no. 1, article 107824, doi:10.1016/j.jpaa.2024.107824, https://doi.org/10.1016/j.jpaa.2024.107824 (preprint: arXiv:2303.02912, https://arxiv.org/abs/2303.02912)
13. Haicheng Zhang, Xinran Zhang and Zhiwei Zhu, A Note on Odd Periodic derived Hall algebras, arXiv:2307.09071, https://arxiv.org/abs/2307.09071 (to appear in Journal of Algebra and Its Applications, doi:10.1142/S0219498825502822, https://doi.org/10.1142/S0219498825502822)
14. Jie Sheng, Xueqing Chen and Fan Xu, On two Hall algebra approaches to odd periodic triangulated categories, Science China Mathematics 61 (2018), no. 2, 317–324, doi:10.1007/s11425-016-9129-4, https://doi.org/10.1007/s11425-016-9129-4
15. Ji Lin and Liangang Peng, Semi-derived Ringel–Hall algebras and Hall algebras of odd-periodic relative derived categories, Science China Mathematics 67 (2024), no. 8, 1735–1760, doi:10.1007/s11425-022-2145-x, https://doi.org/10.1007/s11425-022-2145-x
16. Changjian Fu, Zhanhong Liang and Ming Lu, GIM and Elliptic Lie algebras via Ringel–Hall Lie algebras, arXiv:2608.07877, https://arxiv.org/abs/2608.07877