Homological integral of Hopf algebras

Abstract

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

Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras

“…Here we answer the question for the case G = SL(n, C), giving ν in a form which, we conjecture, remains valid for all semisimple groups G. First we need the following lemma, a straightforward exercise in the use of change of rings formulae proved in [LWZ,Lemma 2.6]. Recall the definition of τ -normal elements in (2.2).…”

“…If A has finitely many 1-dimensional modules, then A ab is finite-dimensional. By [LWZ,Lemma 4.5], io(A) divides dim A ab , which is finite. The second statement follows from (b) and (c).…”

“…We can generalise this result somewhat, as in the following proposition. Recall that io(A) denotes the integral order of A which is (by definition) the order of ξ (see [LWZ,Definition 2.2]). …”

Noetherian Hopf Algebras

“…The case of Gelfand-Kirillov dimension zero can be dismissed, since the only prime finite dimensional Hopf algebra over k is k itself. [21,Section 7]. This was carried forward by Brown and Zhang in [12] under the following hypotheses:…”

On Noetherian affine prime regular Hopf algebras of Gelfand-Kirillov dimension 1

“…The pioneering effort in this direction is [6]. In this nice paper, by using homological properties, the authors define the integrals for a large class of infinite dimensional Hopf algebras as follows.…”