We construct four families of Artin-Schelter regular algebras of global dimension four. Under some generic conditions, this is a complete list of Artin-Schelter regular algebras of global dimension four that are generated by two elements of degree 1. These algebras are also strongly noetherian, Auslander regular and Cohen-Macaulay. One of the main tools is Keller's higher-multiplication theorem on A ∞ -Ext-algebras.

Abstract. The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke's theorem for infinite dimensional Hopf algebras. The generalization of Maschke's theorem and homological integrals are the keys to studying noetherian regular Hopf algebras of Gelfand-Kirillov dimension one. IntroductionLet H be a Hopf algebra over a base field k. Larson and Sweedler's result (GLD) ⇔ (ITG) is a generalization of Maschke's theorem for finite groups. These results are so elegant and useful that we cannot help attempting to extend them to the infinite dimensional case. The extension of (ANT) ⇒ (GLD) is quite successful. In [WZ2, 0.1] the authors proved the following: Suppose that H is a finite module over its affine center and that char k = 0. If S 2 = id H , then H has finite global dimension. It is well-known that the converse of [WZ2, 0

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.

We study a Morita-equivalent version of the Zariski cancellation problem.

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