Connected (graded) Hopf algebras
Connected (graded) Hopf algebras
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DOI：
10.1090/tran/7686
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发表时间：
2016-01
影响因子：
1.3
通讯作者：
K. Brown;P. Gilmartin;James J. Zhang
中科院分区：
文献类型：
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作者：
K. Brown;P. Gilmartin;James J. Zhang
研究了特征为零的代数闭域k上两类无限维Hopf代数的代数性质和同调性质。第一类是连通为分次代数的Hopf k-代数，第二类是连通为余代数的Hopf k-代数。对于这里给出的许多结果，我们假定Hopf代数具有有限的Gel‘fand-Kirillov维。证明了：如果Hopf代数H是有限Gel‘fand-Kirillov维n的连通分次代数，则H是整体维为Cohen-Macaulay，Artin-Schelter正则和Auslander正则的Notherian域，具有S^2=ID_H，是Calabi-Yau.还提供了关于H的Hilbert级数的详细信息。我们的结果保留了第一类代数(适当地)包含在第二类代数中的可能性。对于第二类具有连通余代数的有限Gel‘fand-Kirillov维的Hopf k-代数，证明了其基础余代数是整体维为n的Artin-Schelter正则.这两类Hopf代数与有限维李代数的包络代数有许多共同的特征.例如，这两类中的任何一个代数只有当它是交换多项式代数时才满足多项式恒等式。然而，作为我们的主要结果之一，我们构造了Gel‘fand-Kirillov维为5的Hopf k-代数H的一个例子，它是连通的，分次连通为代数，连通为余代数，但对于任何李代数g，它不同构于U(G)。
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 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.