Higher preprojective algebras, Koszul algebras, and superpotentials

In this article we study higher preprojective algebras, showing that various known results for ordinary preprojective algebras generalize to the higher setting. We first show that the quiver of the higher preprojective algebra is obtained by adding arrows to the quiver of the original algebra, and these arrows can be read off from the last term of the bimodule resolution of the original algebra. In the Koszul case, we are able to obtain the new relations of the higher preprojective algebra by differentiating a superpotential and we show that when our original algebra is $d$-hereditary, all the relations come from the superpotential. We then construct projective resolutions of all simple modules for the higher preprojective algebra of a $d$-hereditary algebra. This allows us to recover various known homological properties of the higher preprojective algebras and to obtain a large class of almost Koszul dual pairs of algebras. We also show that when our original algebra is Koszul there is a natural map from the quadratic dual of the higher preprojective algebra to a graded trivial extension algebra.

Gerstenhaber Bracket on the Hochschild Cohomology via An Arbitrary Resolution

We prove formulas of different types that allow us to calculate the Gerstenhaber bracket on the Hochschild cohomology of an algebra using some arbitrary projective bimodule resolution for it. Using one of these formulas, we give a new short proof of the derived invariance of the Gerstenhaber algebra structure on Hochschild cohomology. We also give some new formulas for the Connes differential on the Hochschild homology that lead to formulas for the Batalin–Vilkovisky (BV) differential on the Hochschild cohomology in the case of symmetric algebras. Finally, we use one of the obtained formulas to provide a full description of the BV structure and, correspondingly, the Gerstenhaber algebra structure on the Hochschild cohomology of a class of symmetric algebras.

HOCHSCHILD (CO)HOMOLOGY OF ℤ2×ℤ2-GALOIS COVERINGS OF QUANTUM EXTERIOR ALGEBRAS

Let Aq=k〈x,y〉/(x2,xy+qyx,y2) be the quantum exterior algebra over a field k with $\mathrm {char}\,k\neq 2$, and let Λq be the ℤ2×ℤ2-Galois covering of Aq. In this paper the minimal projective bimodule resolution of Λq is constructed explicitly, and from it we can calculate the k-dimensions of all Hochschild homology and cohomology groups of Λq. Moreover, the cyclic homology of Λq can be calculated in the case where the underlying field is of characteristic zero.