Connected (graded) Hopf algebras

Abstract

We study algebraic and homological properties of two classes of infinite dimensional Hopf algebras over an algebraically closed field k of characteristic zero. The first class consists of those Hopf k-algebras that are connected graded as algebras, and the second class are those Hopf k-algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel'fand-Kirillov dimension. It is shown that if the Hopf algebra H is a connected graded Hopf algebra of finite Gel'fand-Kirillov dimension n, then H is a noetherian domain which is Cohen-Macaulay, Artin-Schelter regular and Auslander regular of global dimension n. It has S 2 = Id H , and is Calabi-Yau. Detailed information is also provided about the Hilbert series of H. Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf k-algebras of finite Gel'fand-Kirillov dimension n with connected coalgebra, the underlying coalgebra is shown to be Artin-Schelter regular of global dimension n. Both these classes of Hopf algebra share many features in common with enveloping algebras of finite dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf k-algebra H of Gel'fand-Kirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to U (g) for any Lie algebra g.

Unimodular graded Poisson Hopf algebras

“…When working with graded Hopf algebras it is frequently useful to make use of the adjustment permitted by the following lemma, a slight improvement of [, Lemma 2.1(2)]. Lemma Let be a Hopf algebra which is a connected graded algebra, .…”

“…Proof (i)This is [, Lemma 2.1(2)]. (ii)Clearly it is enough to show that for all . So, let and let .…”

“…The possible Hopf structures in this case are as described in the following theorem. Details concerning part (i) can be found at [, Theorem 1]. Its nontrivial content is due to Serre for , (see for example, [, Theorem 6.2.2]), and to Lazard for .…”

“…It should be observed that this example illustrates the fact that a Poisson Hopf polynomial algebra may be a graded Poisson Hopf algebra with respect to more than one grading, and that the hypotheses of the main theorem may apply in a proper subset of the possible cases. Example Let be the connected Hopf algebra of GK‐dimension 5 constructed in [, Theorem 5.6]. That is, with relations given by setting all commutators of the generators equal to 0, except for and .…”

“…A result of states that a Hopf ‐algebra of finite Gelfand–Kirillov dimension which is connected graded as an algebra is Calabi–Yau. One might suspect that there should be a Poisson version of this result, and indeed our main result is the following theorem, whose proof uses this noncommutative result from , applied to the Poisson enveloping algebra of a graded Poisson Hopf algebra. Theorem Let be a Poisson Hopf algebra.…”