Higher Koszul algebras and A-infinity algebras

Abstract

We study a class of A ∞ -algebras, named (2, p)-algebras, which is related to the class of p-homogeneous algebras, especially to the class of p-Koszul algebras. A general method to construct (2, p)-algebras is given. Koszul dual of a connected graded algebra is defined in terms of A ∞ -algebra. It is proved that a p-homogeneous algebra A is p-Koszul if and only if the Koszul dual E(A) is a reduced (2, p)-algebra and generated by E 1 (A). The (2, p)-algebra structure of the Koszul dual E(A) of a p-Koszul algebra A is described explicitly. A necessary and sufficient condition for a p-homogeneous algebra to be a p-Koszul algebra is also given when the higher multiplications on the Koszul dual are ignored.  2005 Elsevier Inc. All rights reserved.

Piecewise-Koszul algebras

“…The multiplications {m n } n 3 are called the higher multiplications of E. An augmented A ∞ -algebra E is an A ∞ -algebra satisfying the following conditions, where for the notions of A ∞ -morphism and strict A ∞ -morphism, we refer to [7] and [12] for the details:…”

“…From [7,9], we know that T (J * ) is an augmented differential bigraded algebra and E(A) = Ext * A (F, F) ∼ = H(T (J * )). By Lemma 3.3, the Koszul dual E(A) of A is an augmented bigraded A ∞ -algebra up to a quasi-isomorphism.…”

“…Many nice homological properties of Koszul algebras have been shown in the research areas of commutative and noncommutative algebras, such as algebraic topology, algebraic geometry, quantum group, and Lie algebra (see [2][3][4], etc.). Thirty years later, motivated by the cubic Artin-Schelter regular algebras, Berger extended the concept to higher homogeneous algebras in [5], one can find more discussions under the name of d-Koszul algebra in [6], or higher Koszul algebra in [7], the latter explained Koszulity by A ∞ -language.…”

“…We also call E(A) the Koszul dual of A, as that other literatures have named. The following results were stated in [4,6,7], respectively:…”

Кошулеподобные Алгебры И Модули

“…Многие авторы предпочитают называть их " -кошулевыми", а не "неквадратичными кошулевыми" (см. [7], [8]), где 2 -фик-сированное целое число. В 2002 г. авторы работы [9], первоначальной целью которых было описание периодических резольвент тривиальных расширений алгебр путей колчана Дынкина в двудольной интерпретации, ввели понятие почти кошулевой алгебры; исследуя условия конечной порожденности Ext-алгебр градуированных алгебр, Грин и Маркос в 2005 г. определили класс -кошулевых алгебр (см.…”