Coassociative Lie Algebras

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.

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“…, 1, 2) is isomorphic as an algebra to such an enveloping algebra. See [Wa3] for the relevant definition and theorem. 5.5.…”

Section: 2mentioning

confidence: 99%

Connected (graded) Hopf algebras

Brown

Gilmartin

Zhang

2018

Trans. Amer. Math. Soc.

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“…For an anti-cocommutative CLA L, the enveloping algebra U (L) is a connected Hopf algebra since L is conilpotent by [WZZ3,Lemma 2.8(b)].…”

Section: Coassociative Lie Algebras and Their Enveloping Algebrasmentioning

confidence: 99%

“…Proof. (a) The subcoalgebra P 2 (H) + k1 is connected and counital by [WZZ3,Lemma 2.4]. Then the subbialgebra of H generated by P 2 (H) + k1 is connected, and whence a connected Hopf algebra [Mo,Lemma 5.2.1].…”

Section: Coassociative Lie Algebras and Their Enveloping Algebrasmentioning

confidence: 99%

“…Let x 3 be a basis element in P 2 (L) and x 4 be a basis element in P 3 (L). Write δ(x 4 ) = f ⊗ x 3 + x 3 ⊗ g + ax 3 ⊗ x 3 with f, g ∈ P 1 (L) and a ∈ k. Coassociativity on x 4 implies that a = 0, f = g = 0 and δ(x 3 ) = bf ⊗ f for some b ∈ k. Thus L is symmetric and hence quasi-equivalent to a Lie algebra by [WZZ3,Corollary 2.6].…”

Section: Coassociative Lie Algebras and Their Enveloping Algebrasmentioning

confidence: 99%

“…The enveloping algebra of the case (e) has an interesting property that S 2 is not the identity, see [WZZ3,Example 4.2].…”

Section: Dimensionmentioning

confidence: 99%

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