The semi-stable ChowHa of a quiver with stability is defined as an analog of the Cohomological Hall algebra of Kontsevich and Soibelman via convolution in equivariant Chow groups of semistable loci in representation varieties of quivers. We prove several structural results on the semistable ChowHa, namely isomorphism of the cycle map, a tensor product decomposition, and a tautological presentation. For symmetric quivers, this leads to an identification of their quantized Donaldson-Thomas invariants with the Chow-Betti numbers of moduli spaces. map for fine moduli spaces of quivers. We exhibit a tensor product decomposition (Theorem 6.1) of the CoHa into all semi-stable CoHa's for various slopes of the stability, categorifying the Harder-Narasimhan, or wall-crossing, formula of [14]. We give a "tautological" presentation of the semi-stable ChowHa (Theorem 8.1), in the spirit of [7], which generalizes the algebraic description of [12] of the CoHa. Quite surprisingly, such a tautological presentation remains valid for the equivariant Chow groups of stable loci in representation varieties (Theorem 9.1). From this, we conclude that the quantized Donaldson-Thomas invariants of a symmetric quiver are given by the Poincaré polynomials of the Chow groups of moduli spaces of stable quiver representations (Theorem 9.2). This shows that quantized Donaldson-Thomas invariants are of algebro-geometric origin; compare [13] where quantized Donaldson-Thomas invariants are interpreted via intersection cohomology of moduli spaces of semi-stable quiver representations.The proofs of these structural results basically only use the Harder-Narasimhan stratification of representation varieties of [14], properties of equivariant Chow groups, and the result of Efimov [4]. All structural results are illustrated by examples in Section 10: we first give a complete description of the Hall algebra of a two-cycle quiver, which is the only symmetric quiver with known representation theory apart from the trivial and the one-loop quiver whose CoHa's are already described in [12]. Then we consider the only non-trivial (i.e. not isomorphic to the CoHa of a trivial quiver) semi-stable ChowHa for the Kronecker quiver -we observe that it is not super-commutative, but still has the same Poincaré-Hilbert series as a free super-commutative algebra; for this, the representation theory of the Kronecker quiver is used essentially. Then we illustrate the calculation of Chow-Betti numbers of moduli spaces of stable representations in the context of classical invariant theory, and finally hint at an algebraic derivation of the explicit formula of [15] for quantized Donaldson-Thomas invariants of multiple loop quivers.The paper is organized as follows: After reviewing basic facts on quiver representations (Section 2) and the definition of quantized Donaldson-Thomas invariants (Section 3), we define the semi-stable ChowHa in Section 4. In Section 5 we study the cycle map from ChowHa to CoHa, and prove it to be an isomorphism via induction over Harder-Narasimhan str...

We prove that Chow groups of certain non-commutative Hilbert schemes have a basis consisting of monomials in Chern classes of the universal bundle. Furthermore, we realize the cohomology of non-commutative Hilbert schemes as a module over the Cohomological Hall algebra.

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A result of A. King and C. Walter asserts that the Chow ring of a fine quiver moduli space is generated by the Chern classes of universal bundles if the quiver is acyclic. We will show that defining relations between these Chern classes arise geometrically as degeneracy loci associated to the universal representation. R and let σ − J :Lemma 3.5. (i) For every J ∈ Φ θ , the cone σ − J is a simplex of dimension ♯J.Proof. (i) We have to show that the elements s * λ α with α ∈ J are linearly independent over the reals. Let us assume there were an element 0as λ + is supported in J. By stability of J c , we obtain that θ, λ > 0. But, on the other hand, we also get χ α , r * (−λ) = 0 and thus, θ, −λ > 0. A contradiction.Proposition 3.7. We obtainand let X := X Σ + be the toric variety associated to Σ + . Then X is an open subset of R which is defined as the union X = J∈Φ θ X σ + J . By definition,and thus, it is clear that R θ = X.toric varieties. This morphism is T + -equivariant via s : T + → T − and therefore, T -invariant with respect to the induced T -actions via r : T → T + . We know that a geometric T -quotient exists. Thus, it suffices to show that Y is a categorical T -quotient. Let f : R θ → Z be a T -invariant morphism of varieties. Let X J := X σ + J . This is an affine open subset of R. Lemma 3.6 shows thatand therefore, η J : X J → Y J := X σ − J is a universal categorical quotient (cf. [16, Theorem 1.1]). Note that Y J is an open subset of Y and η J coincides with the restriction of η. By the quotient property, we obtain that there exists a unique morphism g J :because η Jν is a universal categorical quotient. As g J is uniquely determined by the property g J η J = f J , we obtain g Jν |Y J = g J . This proves that the maps g J with J ∈ Φ θ glue together to a map g : Y → Z with gη = f . Conversely, every such map g has to fulfill g|Y J = g J .We have obtained an explicit description of the fan of M θ . This enables us to prove Proposition 3.2 with the help of Danilov's theorem (cf. [3, Theorem 10.8]).

We show that the cohomology ring of a quiver Grassmannian associated with a rigid quiver representation has property (S): there is no odd cohomology and the cycle map is an isomorphism; moreover, its Chow ring admits explicit generators defined over any field. From this we deduce the polynomial point count property. By restricting the quiver to finite or affine type, we are able to show a much stronger assertion: namely, that a quiver Grassmannian associated with an indecomposable (not necessarily rigid) representation admits a cellular decomposition. As a corollary, we establish a cellular decomposition for quiver Grassmannians associated with representations with rigid regular part. Finally, we study the geometry behind the cluster multiplication formula of Caldero and Keller, providing a new proof of a slightly more general result.