gr.group theory – Is a Hopf algebra a group object of some category?

The page of ncatlab on group object states that:

A group object in $mathrm{CRing}^{mathrm{op}}$ is a commutative Hopf
algebra.

Question: Is a (noncommutative) Hopf algebra a group object of some category?
(let assume finite dimensional, if necessary.)

The page of Wikipedia on group object states that:

Hopf algebras can be seen as a generalization of group objects to
monoidal categories.

It is not clear to me how this sentence answers the above question.

hopf algebras – Do chains send homotopy inverse limits of spaces to homotopy inverse limits of $E_infty$-coalgebras?

Let $X_bullet := … X_2 to X_1$ be a tower of connected and simple spaces
with the following properties:

The induced tower $H_ast(X_bullet; mathbb{F}_p)$ of graded $mathbb{F}_p$-vector spaces
is Mittag-Leffler and lifts to a tower of graded abelian hopf algebras over $mathbb{F}_p$.
The induced tower $pi_1(X_bullet)$ is Mittag-Leffler.

By a theorem of Goerss the canonical morphism $$ H_ast(holim X_bullet; mathbb{F}_p)to lim H_ast(X_bullet; mathbb{F}_p)$$ is an isomorphism, where the limit on the right hand side is taken in the category of graded abelian hopf algebras over $ mathbb{F}_p$.

The limit in the category of graded abelian hopf algebras over $ mathbb{F}_p$ forgets to the limit in the category of graded cocommutative coalgebras over $mathbb{F}_p$ but does generally not forget to the limit in graded $mathbb{F}_p$-vector spaces.

Is there a similar result on the chain level?

Is it true under the assumptions on $X_bullet$ that the canonical morphism $$ C_ast(holim X_bullet; mathbb{F}_p)to holim C_ast(X_bullet; mathbb{F}_p)$$ is a quasi-isomorphism, where
$C_ast(-; mathbb{F}_p)$ are chains with $mathbb{F}_p$-coefficients, and the homotopy limit on the right hand side is taken in the $infty$-category of $E_infty$-coalgebras over $mathbb{F}_p$?

This would follow of course from Goerss theorem if homology $H_ast$ would send
the homotopy limit in $E_infty$-coalgebras (over $mathbb{F}_p$) of the tower $ C_ast(X_bullet; mathbb{F}_p)$ to the limit in graded cocommutative coalgebras over $mathbb{F}_p$.

Does one know such a result?

Can one say more if one additionally assumes that the tower $X_bullet := … X_2 to X_1$
of spaces refines to a tower of grouplike $E_infty$-spaces?

Under this assumption the induced tower $C_ast(X_bullet; mathbb{F}_p)$
is a tower of $E_infty$-hopf algebras over $mathbb{F}_p$, i.e. abelian group objects (in the derived sense) in the $infty$-category of $E_infty$-coalgebras over $mathbb{F}_p.$

Therefore by Goerss theorem the canonical morphism $$ C_ast(holim X_bullet; mathbb{F}_p)to holim C_ast(X_bullet; mathbb{F}_p)$$ would be a quasi-isomorphism if homology $H_ast$ would send
the homotopy limit in $E_infty$-hopf algebras of the tower $ C_ast(X_bullet; mathbb{F}_p)$ to the limit in graded abelian hopf algebras over $mathbb{F}_p$.

Does one know such a result?

Primitive elements in Hopf algebras over the integers

Let $H$ be a Hopf algebra over $mathbb Z$, and assume that $H$ is cocommutative, graded, generated in degree $1$, and connected (its degree-$0$ part is $mathbb Z$).

Are there nice, natural conditions that will enforce that $H$ is a universal enveloping algebra of a Lie algebra over $mathbb Z$?

For example, if $H$ is $mathbb Z$-free, then the Milnor-Moore theorem implies $Hotimesmathbb Q=U(P)$ for $P’$ the space of primitives in $Hotimesmathbb Q$, and presumably $P’=Potimesmathbb Q$ for $P$ the $mathbb Z$-module of primitives in $H$.

I’m sure this works in a much more general setting, but I failed to locate relevant papers or books on Hopf algebras over non-fields.

Note that this question is related to the MO question
Integral Milnor-Moore theorem, though it seems orthogonal.

ct.category theory – Relation Hopf Categories and Categorified Quantum Groups

In the paper Hopf Categories Crane and Frenkel gave a definition of a Hopf category, which they considered as a categorification of a quantum group. Categorifications of quantum groups have later been defined by Rouquier in 2-Kac-Moody groups and Khovanov and Lauda in their series of papers KL1, KL2 and KL3. The latter is in fact a 2-category not a category, where the former seems to be representation categories of something called trialgebras.

Are there any results on the relation of the two notions of categorification of quantum groups?

Hopf “algebroid” structure of a groupoid convolution algebra?

This question is already posted in math.stackexchange, but didn’t receive any answer. I’m not sure if this question fits in here, but surely someone in here can guide me to the correct answer.

To male thinks simple as possible, lets say we have a discrete group $G.$ Then the then the group algebra $mathbb{C}(G)$ (of finitely supported complex valued functions on $G$) has a convolution and an involution operation given by $$(fstar g)(x)=sum_{x=ab}f(a)g(b), qquad f^{ast}(x)=overline{f(x^{-1})}$$

It is easier to interpret $mathbb{C}(G)$ as the free complex vector space spanned by $G$ for notational convenience. I came across a statement that says this $ast$-convolution algebra has a natural Hopf algebra structure given by comultiplication $Delta(g)=gotimes g$ and counit $epsilon(g)=1,$ then extended linearly. Also antipode is given by $ast$-operation extended antilinearly.

Now I would like to know, what happen if we replace the group $G$ with a groupoid? My naïve guess is that we would get a Hopf algebroid (many object analogue of the known construction). If it is the case, how would the coalgebra look like? Can anyone explain me this structure or direct me to a (simple) reference?

reference request – Is the kernel of an action of a Hopf algebra on an algebra a biideal?

I think, this must be simple, but I am not a specialist in this field, so excuse me. I asked this a week ago at MSE, but without success.

S.Dascalescu, C.Nastasescu and S.Raianu define the action of a Hopf algebra $H$ on an (associative) algebra $A$ as a map $Htimes Aowns (h,a)mapsto hcdot ain A$ which

  • is an action of $H$ on $A$ as an algebra on a vector space, and
  • satisfies two supplementary conditions:
    $$
    hcdot(acdot b)=sum(h_1cdot a)cdot (h_2cdot b),qquad
    hcdot 1_A=varepsilon(h)cdot 1_A.
    $$

I think that the kernel of such an action, i.e. the set
$$
I={hin H:quad forall ain Aquad hcdot a=0},
$$

must be (not only a two-sided ideal in $H$ as in an algebra, but also) a biideal in $H$ as in a bialgebra, i.e.
$$
Delta(I)subseteq Iotimes H+Hotimes I.
$$

Is this true?

soft question – References for Hopf Galois module theory

I am a PhD student and I am really interested in Galois module theory, both in a “classical” and in a “nonclassical” sense. In the last months I have been reading about Hopf Galois theory, since it seems to be a nice way to find the structure of the ring of integers (or valuation ring, in the local case), when the classical approach fails.

In particular, I am reading Childs’s book “Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory”, where he makes a nice exposition of the known facts untile 2000.

I was looking for some more recent results, but my search was not really fruitful. Could you please advise some more modern papers (or books, or anything you want) discussing about Hopf Galois module theory, so Galois module theory applied in the nonclassical setting of Hopf Galois extensions?

at.algebraic topology – Hopf invariants of elements from spherical fibrations

Let $G_n$ be the space of homotopy-equivalences of $S^{n-1}$. Evaluation produces a map $G_{n} to S^{n-1}$. For $n = 2m+1$, I would like to understand the induced map on $pi_{4m-1}$. More precisely, what I would really like to know are the Hopf invariants of elements in the image of $pi_{4m-1} G_{2m+1} to pi_{4m-1} S^{2m}$.

For $m = 1$, the classical Hopf fibration gives rise to a generator of $pi_3 S^2 = bf Z$, and this element even lies in the image of $pi_3 SO(3) to pi_3S^2$ which factors through $pi_3 G_3 to pi_3 S^2$.

For $m = 2$, I do not know the answer, but I can prove that every map lying in the image of $pi_7 S0(5) to pi_7S^4 cong {bf Z} oplus {bf Z}/12bf Z$ has Hopf invariant divisible by $12$. Moreover, the question whether $1$ is attained might be equivalent to asking whether ${bf H}P^3$ is homotopy equivalent to the total space of a fibration with fiber $S^4$ and base $S^8$.

I do not know what happens for $m geq 3$. The case $m = 4$, where $pi_{15} S^8$ contains an element of Hopf invariant 1, would be especially interesting. Maybe there is also a relation to the non-existence of ${bf O}P^3$?

reference request – Definition of Braided dual of a Hopf Algebra

I have been reading the paper “Integrating Quantum Groups over Surfaces” by Jordan, Brochier and Ben-Zvi.

At the page 42 they talk about the braided dual algebra $tilde{H}$ to a quasi-triangular Hopf algebra $H$ but I can’t find a definition of this, even in the references they give in the paper.

Any reference to this would be welcomed.

gr.group theory – Are the symmetric groups integrable as Hopf algebras?

Let $G$ be a group. For $g,h in G$, let $(g,h)=g^{-1}h^{-1}gh$ be a commutator. The normal subgroup $G’ = langle (g,h) | g,h in G rangle$ is called the commutator subgroup or derived subgroup.

An integral of $G$ is a group $H$ such that $H’simeq G$. The problem of the existence of an integral was first mention by B.H. Neumann is this paper (1956). A group without integral is called non-integrable. The smallest non-integrable finite group is the symmetric group $S_3$; moreover $S_n$ is non-integrable $forall n ge 3$.

Here are two recent references about integrals of groups: Filom-Miraftab (2017) and Araújo-Cameron-Casolo-Matucci (2019).

The commutator subgroup is the smallest normal subgroup for which the quotient is commutative. This notion was extended to semisimple Hopf algebra (Burciu, 2012) and is called commutator subalgebra. It is the smallest normal left coideal subalgebra for which the quotient is commutative. Then let call a semisimple Hopf algebra integrable if it is isomorphic to the commutator subalgebra of a semisimple Hopf algebra.

Question: Are the Hopf algebras $mathbb{C}S_n$ integrable? What if $n=3$?

More generally we can ask whether there exist a non-integrable finite group which is integrable as Hopf algebra, and if so, whether there is one which is not, and if no, whether every finite dimensional semisimple Hopf algebra is integrable.