Hopf algebras with rigid dualizing complexes

Abstract

Let H be a Hopf algebra over a base field. If H has an N-filtration such that the associated graded ring is connected graded noetherian and has enough normal elements, then H is Gorenstein. This gives a partial solution to a question of Brown and Brown-Goodearl. As a consequence, every quotient Hopf algebra of a generic quantized coordinate ring of a connected semisimple Lie group is Auslander-Gorenstein and Cohen-Macaulay. The last statement answers a question of Goodearl-Zhang.

Noetherian Hopf Algebras

“…On the other hand, O(SL 2 (k)) is not pointed, since it has a four-dimensional simple subcoalgebra, namely the subspace spanned by the matrix entry functions X 11 , X 12 , X 21 , X 22 . Since examples of this type only seem to appear in dimension 3 and higher, Brown and Zhang raised the following question [ The most useful condition implying pointedness is the existence of enough grouplike and skew primitive elements, as the following lemma shows.…”

Calabi-Yau pointed Hopf algebras of finite Cartan type

“…for any a ⊗ b ∈ R e and (c ⊗ d)#g ∈ R e # Γ. Then each Hom R e (P i , R e ) is an R e # Γ-module as well: (16) […”

“…where the R e # Γ-module structure on Hom R e (P, R e ) is defined in (16). Now the canonical isomorphism from Hom R e (P, R e ) ⊗ W to Hom R e (P, R e ⊗ W ) is a Γ-A e -bimodule isomorphism.…”

“…AS-Gorenstein Hopf algebras have been recently intensively studied (e.g. [6,7,15,16,24,25]). One of important properties of an AS-Gorenstein Hopf algebra is the existence of a homological integral, which generalizes Sweedler's classical integral of a finite dimensional Hopf algebra cf.…”

Noetherian Hopf algebras

“…Detailed proofs for classes 1, 2 and 4(i) can be found in [BZ,§6]; see [GZ] for class 4(ii). The proof for class (2.2)6 given in [LWZ2] is different in flavour; we discuss it briefly in Remark (5.2)(b). The most striking of these positive cases for Question E is class (2.2)5, affine noetherian Hopf algebras satisfying a polynomial identity -the result is a theorem of Wu and Zhang which is both beautiful and technical.…”