PI Hopf algebras of prime characteristic.
The author studies a condition when a cocommutative Hopf algebra \(H\) of prime characteristic has a polynomial identity (as an algebra). The case of a semidirect product \(H=u(L)\#K[G]\), where \(u(L)\) is a restricted enveloping algebra and \(K[G]\) is a group algebra, \(G\) acting on \(L\) by automorphisms, was settled by \textit{Yu. A. Bakhturin} and \textit{V. M. Petrogradsky} [in J. Lie Theory 12, No. 2, 369-395 (2002; Zbl 1014.17006)]. In some sense, this paper studies an opposite case. A bialgebra is called `hyperalgebra' if it is connected and cocommutative. Theorem. Let \(H\) be a hyperalgebra over a perfect field \(K\). If \(\text{char\,} K=p>0\), then it is also assumed that \(H\) is reduced. Then \(H\) is a PI-algebra iff \(H\) is commutative. Next, the author studies smash products and suggests the Conjecture: Let \(H\) be a hyperalgebra over a perfect field \(K\). If \(\text{char\,}K>0\), assume also that \(H\) is reduced. Let \(G\) be a group acting on \(H\) by bialgebra automorphisms. Then the smash product \(H\#K[G]\) is PI iff (1) \(H\) is commutative; (2) there exist normal subgroups \(G_0\subset G_1\subset G\) such that \(G/G_1\) is finite, \(G_1/G_0\) is Abelian, and \(G_0\) is a finite \(p\)-group if \(\text{char\,}K=p>0\) and trivial if \(\text{char\,}K=0\). (3) \(G_1\) acts trivially on \(H\). The author checks sufficiency. The necessity of conditions (1), (2) follows from respective results of D. S. Passman, Yu. A. Bakhturin and V. M. Petrogradsky. The author presents some positive evidence that this conjecture should be true by proving it for two particular cases.
scientific article; zbMATH DE number 1924691