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.

在本文中，我们研究了较高的投影前代数，表明普通投影前代数的各种已知结果都可以推广到较高的设置。我们首先表明，通过将箭头添加到原始代数的颤动中可以获得较高的预投影代数的颤动，并且这些箭头可以从原始代数的双模分解的最后一项中读出。在Koszul案例中，我们能够通过区分超势来获得更高的投影前代数的新关系，并且我们证明了当我们原始的代数是$ d $-遗传时，所有关系都来自超势。然后，我们为$ d $的较高预投影代数构造所有简单模块的投影分辨率。遗传代数。这使我们能够恢复高射影前代数的各种已知同源性，并获得一大类几乎是Koszul对偶的代数。我们还表明，当我们的原始代数是Koszul时，存在一个自然映射，从高阶投影前代数的二次对偶到渐变平凡扩展代数。