An Application of a Theorem of Sheila Brenner for Hochschild Extension Algebras of a Truncated Quiver Algebra

Let A be a truncated quiver algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of A is zero in A. We give the number of the indecomposable direct summands of the middle term of an almost split sequence for a class of Hochschild extension algebras of A by the standard duality module D(A).

     

The ordinary quivers of Hochschild extension algebras for self-injective Nakayama algebras

Let T be a Hochschild extension algebra of a finite dimensional algebra A over a field K by the standard duality A-bimodule Hom_K(A, K). In this paper, we determine the ordinary quiver of T if A is a self-injective Nakayama algebra by means of the ?-graded second Hochschild homology group HH₂(A) in the sense of Sköldberg.     

One-pot reactions of bicyclic zinc enolate generated from Ni-catalyzed reductive cyclization to furnish octahydro-4,7-ethanobenzofuran-9-one derivatives

The one-pot reactions of catalytically generated bicyclic zinc enolate with various electrophiles are reported. The zinc enolate as a key intermediate is efficiently delivered from Ni-catalyzed reductive cyclization of alkynyl cyclohexadienone. Employing aldehydes, imine, nitroalkene, and α,β-unsaturated carbonyl compounds as electrophiles, this new class of one-pot reactions gave multi-functionalized cis-hydrobenzofurans and octahydro-4,7-ethanobenzofuran-9-one derivatives in moderate to good yields.