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Let \pi \colon Y\to X denote the canonical resolution of the two dimensional Kleinian singularity X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of \omega-semistable properly supported sheaves on Y with fixed slope \mu and \zeta-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on \zeta depending on the polarization \omega and \mu. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of \omega-semistable properly supported sheaves on Y with fixed slope \mu: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams. [ABSTRACT FROM AUTHOR]

Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz (1991) and was further developed in the work of S. Ault (2010). In this paper, we show that, for algebras defined over a field of characteristic 0, the symmetric homology is naturally equivalent to the (one-dimensional) representation homology introduced by the authors in joint work with G. Khachatryan (2013). Using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz, including their main conjecture (2007) on topological interpretation of symmetric homology of polynomial algebras. [ABSTRACT FROM AUTHOR]