Lower bounds of growth of Hopf algebras. (Q2847129)

A famous result of Gromov states that a finitely generated group has polynomial growth, or equivalently, the respective group algebra has finite Gelfand-Kirillov dimension if and only if it has a nilpotent subgroup a finite index. The author poses a similar question: what are necessary and sufficient conditions on a finitely generated Hopf algebra \(H\) such that its Gelfand-Kirillov dimension is finite?NEWLINENEWLINE Let \(H\) be a Hopf algebra over \(k\). A nonzero element \(y\in H\) is called \((1,g)\)-skew primitive if \(\Delta(y)=y\otimes 1+g\otimes y\), and such a \(g\) is called weight of \(y\) and denoted by \(\mu(y)\). Let \(G(H)\) denote the group of group-like elements of \(H\) and let \(C_0=kG(H)\).NEWLINENEWLINE Theorem (first lower bound theorem). Let \(D\supseteq C_0\) be a Hopf subalgebra of \(H\). Let \(\{y_i\}_{i=1}^\omega\) be a set of skew primitive elements such that a) \(\{y_i\}_{i=1}^\omega\) is linearly independent in \(H/D\), b) for all \(i\leq j\), \(y_i\mu(y_j)=\lambda_{ij}\mu(y_j)y_i\) for some \(\lambda_{ij}\in k^*\), c) for each \(i\), \(\lambda_{ii}\) is either 1 or not a root of unity. Then \(\roman{GKdim\,}H\geq\roman{GKdim\,}D+w\).NEWLINENEWLINE The author obtains more lower bounds on the Gelfand-Kirillov dimension of Hopf algebras.