274 research outputs found
Orders of Nikshych's Hopf algebra
Let be an odd prime number and a number field having a primitive
-th root of unity We prove that Nikshych's non-group theoretical
Hopf algebra , which is defined over , admits a Hopf
order over the ring of integers if and only if there is an
ideal of such that . This condition does
not hold in a cyclotomic field. Hence this gives an example of a semisimple
Hopf algebra over a number field not admitting a Hopf order over any cyclotomic
ring of integers. Moreover, we show that, when a Hopf order over
exists, it is unique and we describe it explicitly.Comment: 33 pages. Major changes in the presentatio
On two finiteness conditions for Hopf algebras with nonzero integral
A Hopf algebra is co-Frobenius when it has a nonzero integral. It is proved
that the composition length of the indecomposable injective comodules over a
co-Frobenius Hopf algebra is bounded. As a consequence, the coradical
filtration of a co-Frobenius Hopf algebra is finite; this confirms a conjecture
by Sorin D\u{a}sc\u{a}lescu and the first author. The proof is of categorical
nature and the same result is obtained for Frobenius tensor categories of
subexponential growth. A family of co-Frobenius Hopf algebras that are not of
finite type over their Hopf socles is constructed, answering so in the negative
another question by the same authors.Comment: Minor changes. Final version, to appear in Ann. Sc. Norm. Super. Pisa
Cl. Sci. (5); 33 page
Semisimple Hopf actions on Weyl algebras
We study actions of semisimple Hopf algebras H on Weyl algebras A over a
field of characteristic zero. We show that the action of H on A must factor
through a group algebra; in other words, if H acts inner faithfully on A, then
H is cocommutative. The techniques used include reduction modulo a prime number
and the study of semisimple cosemisimple Hopf actions on division algebras.Comment: v2: 9 pages. To appear in Adv. Mat
On the Hopf-Schur group of a field
Let k be any field. We consider the Hopf-Schur group of k, defined as the
subgroup of the Brauer group of k consisting of classes that may be represented
by homomorphic images of Hopf algebras over k. We show here that twisted group
algebras and abelian extensions of k are quotients of cocommutative and
commutative Hopf algebras over k, respectively. As a consequence we prove that
any tensor product of cyclic algebras over k is a quotient of a Hopf algebra
over k, revealing so that the Hopf-Schur group can be much larger than the
Schur group of k.Comment: 12 pages, latex fil
Extending lazy 2-cocycles on Hopf algebras and lifting projective representations afforded by them
AbstractWe study some problems related to lazy 2-cocycles, such as: extension of (lazy) 2-cocycles to a Drinfeld double and to a Radford biproduct, YetterâDrinfeld data obtained from lazy 2-cocycles, lifting of projective representations afforded by lazy 2-cocycles
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