Connected Hopf algebras of Gelfand-Kirillov dimension four

Classification of connected Hopf algebras of dimension p3 I

“…In this section, we want to compare the algebra structure of (B2) type in Theorem 1.3 with restricted enveloping algebras. The motivation comes from the result by D.-G. Wang, J.J. Zhang, and G. Zhuang [21] on connected affine Hopf algebras in characteristic zero described in the Introduction. Note that restricted enveloping algebras of dimension p 3 are classified in Theorem 1.4 as Hopf algebras of type C. For convenience, we denote by A the algebra described as (B2), namely A is the quotient of the free algebra generated by three variables x, y, z by the ideal generated by: […”

“…According to [8,12,21,26] with respect to algebra structures, up to GK-dimension 4, affine connected Hopf algebras are all isomorphic to universal enveloping algebras. Moreover, D.-G. Wang, J.J. Zhang, and G. Zhuang in [21] studied the algebra structures of connected Hopf algebras over an algebraically closed field of characteristic 0.…”

“…According to [8,12,21,26] with respect to algebra structures, up to GK-dimension 4, affine connected Hopf algebras are all isomorphic to universal enveloping algebras. Moreover, D.-G. Wang, J.J. Zhang, and G. Zhuang in [21] studied the algebra structures of connected Hopf algebras over an algebraically closed field of characteristic 0. Under the assumption that when the GK-dimension equals the dimension of the primitive space plus one, it is proved that the algebra structure of Hopf algebra is always isomorphic to some universal enveloping algebra; see [21, Theorem 0.5 or Theorem 2.7].…”

Coassociative Lie Algebras

“…Theorem 0.2. [WZZ,Theorem 0.3 and Remark 0.4] Suppose that k is of characteristic zero. Let H be a connected Hopf algebra of GK-dimension four.…”

“…A result of Milnor-Moore-Cartier-Kostant [Mo,Theorem 5.6.5] states that any cocommutative connected Hopf algebra over a field of characteristic zero is isomorphic to U (g) for some Lie algebra g. This applies to case (a) of Theorem 0.2. However, the Hopf algebras in Theorem 0.2(b,c) are not cocommutative, hence not isomorphic to U (g) for a usual Lie algebra g. A generalization of Theorem 0.2(b) states that if p(H) = GKdim H − 1, then H is isomorphic to U (L) for some coassociative Lie algebra L [WZZ,Theorem 0.5].…”