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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 Du{a}scu{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.
A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra $B$ in
The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschkes theorem for infinite dimensional Hopf algebras. The generalization of Masc
We show that the definition of unrolled Hopf algebras can be naturally extended to the Nichols algebra $mathcal{B}$ of a Yetter-Drinfeld module $V$ on which a Lie algebra $mathfrak g$ acts by biderivations. Specializing to Nichols algebras of diagonal type, we find unroll
We prove a 20-year-old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two invariants can be
Let $p$ be a prime number and $q=p^m$, with $m geq 1$ if $p eq 2,3$ and $m>1$ otherwise. Let $Omega$ be any non-trivial twist for the complex group algebra of $mathbf{PSL}_2(q)$ arising from a $2$-cocycle on an abelian subgroup of $mathbf{PSL}_2(q)$