This paper investigates a universal PBW-basis and a minimal set of generators for the Hall algebra $\mathscr{H}\left(C_2\left(𝒫\right)\right)$, where $C_2\left(𝒫\right)$ is the category of morphisms between projective objects in a finitary hereditary exact category $𝒜$. When $𝒜$ is the representation category of a Dynkin quiver, we develop multiplication formulas for the degenerate Hall Lie algebra $ℒ$, which is spanned by isoclasses of indecomposable objects in $C_2\left(𝒫\right)$. As applications, we demonstrate that $ℒ$ contains a Lie subalgebra isomorphic to the central extension...

The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras...

We determine the Hochschild cohomology of all finite-dimensional generalized multicoil algebras over an algebraically closed field, which are the algebras for which the Auslander-Reiten quiver admits a separating family of almost cyclic coherent components. In particular, the analytically rigid generalized multicoil algebras are described.

We consider the socle deformations arising from formal deformations of a class of Koszul self-injective special biserial algebras which occur in the study of the Drinfeld double of the generalized Taft algebras. We show, for these deformations, that the Hochschild cohomology ring modulo nilpotence is a finitely generated commutative algebra of Krull dimension 2.