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In this paper, we provide a way of attaching to a torsion pair $(T,F)$ on theheart of a stable $\infty$-category $C$ a cohomological (K-theoretical,categorified) Hall algebra and corresponding left and right representations.More precisely, the algebra is associated to the torsion part, while therepresentation is associated to the torsion-free part.  The left and right actions enable us to construct canonical subalgebras ofthe endomorphism ring of the Borel-Moore homology and K-theory of the modulistack of torsion-free objects, whose "positive parts" recover the cohomologicalHall algebra and the K-theoretical Hall algebra associated to the torsion part$T$, respectively. This provides a new direction that might lead to overcomethe long-standing limitation of the theory of cohomological Hall algebras tojust produce "positive parts" of whole algebras. We also provide a geometricsufficient criterion ensuring the vanishing of the commutator between twodifferent operators.  In the quiver case, we obtain the action of the two-dimensional cohomologicalHall algebra of a quiver on the cohomology of Nakajima quiver varieties withinour framework. Besides the quiver case, we also apply our framework to twotorsion pairs on a smooth projective complex surface, and we investigate thecorresponding Hall algebras and their representations associated to them.Finally, we slightly modify our method to construct representations of thecohomological Hall algebra of zero-dimensional sheaves on $S$ on theBorel-Moore homology of the moduli spaces of Pandharipande-Thomas stable pairson surfaces and on relative Hilbert schemes of points (and we obtain similarresults at the level of K-theory and bounded derived category).

Cohomological Hall algebras and their representations via torsion pairs