Coverings of graded pointed Hopf algebras

We introduce the concept of a covering of a graded pointed Hopf algebra. The theory developed shows that coverings of a bosonized Nichols algebra can be concretely expressed by biproducts using a quotient of the universal coalgebra covering group of the Nichols algebra. If there are enough quadratic relations, the universal coalgebra covering is given by the bosonization by the enveloping group of the underlying rack.Comment: to appear in J. of Algebr

Centers of generic algebras with involution

AbstractLet F be an infinite field of characteristic different from 2. Let n be a positive integer, and let V=Mn(F)⊕Mn(F). The projective symplectic and orthogonal groups, PSpn and POn, act on V by simultaneous conjugation. Results of Procesi and Rowen have shown that F(V)PSpn and F(V)POn are the centers of the generic division algebras with symplectic and orthogonal involutions, respectively. Saltman has shown that F(V)PSpn and F(V)POn are stably isomorphic over F for all n even, and that for all n odd F(V)POn is stably rational over F. Saltman has also shown that for all n for which the highest power of 2 dividing n is less than 8, F(V)PSpn and therefore F(V)POn are stably rational over F. We show that the result is also true for all n for which the highest power of 2 dividing n is 8

The center of the generic division ring and twisted multiplicative group actions

AbstractLet F be a field and let p be a prime. The problem we study is whether the center, Cp, of the division ring of p×p generic matrices is stably rational over F. Given a finite group G and a ZG-lattice, we let F(M) be the quotient field of the group algebra of the abelian group M. Procesi and Formanek [Linear Multilinear Algebra 7 (1979) 203–212] have shown that for all n there is a ZSn-lattice, Gn, such that Cn is stably isomorphic to the fixed field under the action of Sn of F(Gn). Let H be a p-Sylow subgroup of Sp. Let A be the root lattice, and let L=F(ZSp/H). We show that there exists an action of Sp on L(ZSP⊗ZHA), twisted by an element α∈Ext1Sp(ZSp⊗ZHA,L∗), such that Lα(ZSp⊗ZHA)Sp is stably isomorphic to Cp. The extension α corresponds to an element of the relative Brauer group of L over LH. Since ZSp⊗ZHA and ZSp/H are quasi-permutations, L(ZSp⊗ZHA)Sp is stably rational over F. However, it is not known whether Lα(ZSp⊗ZHA)Sp is stably rational over F. Thus the result represents a reduction on the problem since ZSp⊗ZHA is quasi-permutation; however, the twist introduces a new level of complexity