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Register a new userOn two finiteness conditions for Hopf algebras with nonzero integral
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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.
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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 a braided monoidal category $C$, and under certain assumptions on the braiding (fulfilled if $C$ is symmetric), we construct a sequence for the Brauer group $BM(C;B)$ of $B$-module algebras, generalizing Beatties one. It allows one to prove that $BM(C;B) cong Br(C) times Gal(C;B),$ where $Br(C)$ is the Brauer group of $C$ and $Gal(C;B)$ the group of $B$-Galois objects. We also show that $BM(C;B)$ contains a subgroup isomorphic to $Br(C) times Hc(C;B,I),$ where $Hc(C;B,I)$ is the second Sweedler cohomology group of $B$ with values in the unit object $I$ of $C$. These results are applied to the Brauer group of a quasi-triangular Hopf algebra that is a Radford biproduct $B times H$, where $H$ is a usual Hopf algebra over a field $K$, the Hopf subalgebra generated by the quasi-triangular structure $R$ is contained in $H$ and $B$ is a Hopf algebra in the category ${}_HM$ of left $H$-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that $BM(K,H,R) times Hc({}_HM;B,K)$ is a subgroup of the Brauer group $BM(K,B times H,R),$ confirming the suspicion that a certain cohomology group of $B times H$ (second lazy cohomology group was conjectured) embeds into $BM(K,B times H,R).$ New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.
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 Maschkes theorem and homological integrals are the keys to study noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.
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 viewed as a non-semisimple generalization of the Turaev-Viro-Barrett-Westbury $(text{TVBW})$ invariant and the Witten-Reshetikhin-Turaev $(text{WRT})$ invariant, respectively. By a classical result relating $text{TVBW}$ and $text{WRT}$, it follows that the Kuperberg invariant for a semisimple Hopf algebra is equal to the Hennings-Kauffman-Radford invariant for the Drinfeld double of the Hopf algebra. However, whether the relation holds for non-semisimple Hopf algebras has remained open, partly because the introduction of framings in this case makes the Kuperberg invariant significantly more complicated to handle. We give an affirmative answer to this question. An important ingredient in the proof involves using a special Heegaard diagram in which one family of circles gives the surgery link of the three manifold represented by the Heegaard diagram.
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 $mathbf{PSL}_2(q)$. We show that the twisted Hopf algebra $(mathbb{C} mathbf{PSL}_2(q))_{Omega}$ does not admit a Hopf order over any number ring. The same conclusion is proved for the Suzuki group $^2!B_2(q)$ and $mathbf{SL}_3(p)$ when the twist stems from an abelian $p$-subgroup. This supplies new families of complex semisimple (and simple) Hopf algebras that do not admit a Hopf order over any number ring. The strategy of the proof is formulated in a general framework that includes the finite simple groups of Lie type. As an application, we combine our results with two theorems of Thompson and Barry and Ward on minimal simple groups to establish that for any finite non-abelian simple group $G$ there is a twist $Omega$ for $mathbb{C} G$, arising from a $2$-cocycle on an abelian subgroup of $G$, such that $(mathbb{C} G)_{Omega}$ does not admit a Hopf order over any number ring. This partially answers in the negative a question posed by Meir and the second author.
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