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...

Zhou and Zhu have shown that if $𝒞$ is an $(n+2)$-angulated category and $𝒳$ is a cluster tilting subcategory of $𝒞$, then the quotient category $𝒞/𝒳$ is an $n$-abelian category. We show that if $𝒞$ has Auslander-Reiten $(n+2)$-angles, then $𝒞/𝒳$ has Auslander-Reiten $n$-exact sequences.

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.

The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let ${\Lambda }_t$ be the Yoneda algebra of a reconstruction algebra of type ${𝐀}_1$ over a field $.Inthispaper,aminimalprojectivebimoduleresolutionof$t$isconstructed,andthe$-dimensions of all Hochschild homology and cohomology groups of ${\Lambda }_t$ are calculated explicitly.

Let $A$ be a CM-finite Artin algebra with a Gorenstein-Auslander generator $E$, $M$ be a Gorenstein projective $A$-module and $B={\mathrm{End}}_{A}M$. We give an upper bound for the finitistic dimension of $B$ in terms of homological data of $M$. Furthermore, if $A$ is $n$-Gorenstein for $2\le n<\infty$, then we show the global dimension of $B$ is less than or equal to $n$ plus the $B$-projective dimension of ${\mathrm{Hom}}_A(M,E).$ As an application, the global dimension of ${\mathrm{End}}_{A}E$ is less than or equal to $n$.