22 votes
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Open problems in Hopf algebras

There had been a workshop on Hopf algebras and related areas in September 2015. Its report (https://www.birs.ca/workshops//2015/15w5053/report15w5053.pdf) includes a large list of open problems and ...
Todd Leason 's user avatar
19 votes

What is known about the category of Hopf algebras?

The category of Hopf algebras has both kernels and cokernels, and indeed it has all limits and colimits. In fact, quite remarkably, it is locally presentable. This is remarkable because the "easy&...
Theo Johnson-Freyd's user avatar
15 votes
Accepted

Lang's Jacobian identity: slicker, elementary proof?

Awesome question! I haven't looked at Lang's paper yet, so I can't comment on whether this will be a different approach, but it is elementary. I will make use of Glynn's determinant formula at some ...
Gjergji Zaimi's user avatar
15 votes

What is known about the category of Hopf algebras?

The analogy between cocommutative Hopf algebras and groups is remarkably close. The category of (graded) cocommutative $k$-coalgebras is cartesian monoidal under the tensor product and the groups in ...
Peter May's user avatar
  • 30.2k
14 votes

Open problems in Hopf algebras

Let $H$ be a finite dimensional Hopf algebra over a field $k$ of positive characteristic. The following is an important open problem: Is the cohomology ring $Ext^\ast_H(k,k)$ a finitely generated $...
Todd Leason 's user avatar
13 votes

Can one define quantized universal enveloping algebras in a basis-free way?

For complex simple $\mathfrak g$, Drinfeld (1986, p. 807) already characterized his $\mathrm U_h\mathfrak g$ as the unique (up to equivalence and change of parameter) deformation of $\mathrm U\...
Francois Ziegler's user avatar
13 votes

Axiomatic definition of quantum groups

I would have liked to write this as a comment, but with my points tally I can not. So writing this as an answer. In quantum groups, we are probably at a stage group theory was, say in the first half ...
akp's user avatar
  • 311
12 votes
Accepted

Hopf dual of the Hopf dual

I am going to give three counterexamples to your first question. (The third counterexample is courtesy of @Adrien, who did most of the job.) While none of them leads to a full answer of your second ...
darij grinberg's user avatar
12 votes
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Is a Hopf algebra a group object of some category?

Not with their definition, where they assume the underlying category to be cartesian. You can define a notion of "Hopf object" in arbitrary symmetric monoidal categories, where you also need ...
Adrien's user avatar
  • 8,234
12 votes
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Hopf algebra with a non-invertible antipode

Theorem of Takeuchi (in Free Hopf algebras generated by coalgebras, 1971) asserts that free Hopf algebra $H(C)$ over a coalgebra $C$ has injective antipode, and it is bijective precisely (at least ...
Denis T's user avatar
  • 4,416
11 votes

Stable homotopy type theory?

With respect to the first question, expanding on my comment which pointed out the nLab page dependent linear type theory and the article by Urs Schreiber, 'Quantization via Linear homotopy types', I'd ...
David Corfield's user avatar
11 votes
Accepted

An inner product approach to Hopf algebras

This doesn't directly answer your question concerning Hopf structures on $\mathbb{C}^n$, but a particularly well-studied class of Hopf algebras for which the product is the adjoint of the coproduct ...
Mark Penney's user avatar
11 votes
Accepted

The tensor product of two monoidal categories

The book Tensor Categories discusses, with many variations, the details of Robert McRae's answer. Just like for vector spaces, there are a number of related but inequivalent "tensor products" of ...
Theo Johnson-Freyd's user avatar
11 votes

Hopf structure on the universal enveloping of a super Lie algebra

This is true. In other language, if I understand rightly, a super Lie algebra is just a graded Lie algebra with grading over {0,1} (even and odd), with the standard sign conventions as in algebraic ...
Peter May's user avatar
  • 30.2k
11 votes
Accepted

W H Lin's thesis and Hopf subalgebras of the Steenrod algebra

Using @CarloBeenakker's answer, our librarian found an electronic version, produced from the microfilm copy of the original: https://search.proquest.com/docview/302701183 (full text may require ...
John Palmieri's user avatar
11 votes
Accepted

Hopf algebra with a non-grouplike invertible element

Let $L$ be a finite-dimensional $p$-nilpotent restricted Lie algebra over a field of characteristic $p>0$ and consider its restricted enveloping algebra $u(L)$. Then the only group-like element of ...
Salvatore Siciliano's user avatar
10 votes
Accepted

Classification of plethories over $\mathbb{Q}$

The preprint https://arxiv.org/abs/1701.01314 of Magnus Carlson, "Classification of plethories in characteristic zero" answers the question about plethories over $\mathbb{Q}$ in the affirmative.
Charles Rezk's user avatar
  • 26.6k
10 votes
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Algebra in a category

What you are talking about is the notion of monoid in a monoidal category. To show $A$ is a monoid ('algebra'), you need to construct a multiplication map $\mu: A \times A \to A$, that is associative, ...
Jacob White's user avatar
10 votes
Accepted

What kind of algebraic object is $\mathcal{D}_X$? (algebra of diifferential operators). What's special about modules over it?

Proposition 1.2.9 of http://math.columbia.edu/~scautis/dmodules/hottaetal.pdf explains that if $M$ and $N$ are both left $D$-modules and $M'$ and $N'$ are both right $D$-modules then (a) $M\otimes_{R}...
Simon Wadsley's user avatar
10 votes
Accepted

Cartier-Kostant-Milnor-Moore theorem

When $k$ fails to be algebraically closed the theorem is false but the discrepancy can be understood in terms of Galois descent and so in principle understood in terms of Galois cohomology. Suppose $...
Qiaochu Yuan's user avatar
10 votes
Accepted

Name for the action of a bialgebra on an algebra

According to nLab, such an action is called a Hopf action and your data specify a left $B$-module algebra. Such a structure is also referred to in the literature as an algebra in the category (of left ...
Konstantinos Kanakoglou's user avatar
10 votes
Accepted

"Free" Hopf algebra

1) Yes, free means free as an algebra. (I don't think it could possibly mean anything else. "Free Hopf algebras" and "free coalgebras" as left adjoints to the forgetful functor don't exist.) 2) is ...
Najib Idrissi's user avatar
10 votes
Accepted

Subalgebra of a group algebra

The characteristic of the field is important here, when considering Hopf sub-algebras. The Cartier-Kostant-Milnor-Moore theorem says that a cocommutative Hopf algebra $H$ over an algebraically closed ...
Oeyvind Solberg's user avatar
10 votes

Hopf algebra with a non-grouplike invertible element

Let $G$ be any finite group. Then the group algebra $\mathbb{C}[G]$ is, as an algebra, isomorphic to $\bigoplus_V \mathrm{End}(V)$, where the direct sum is over irreducible representations of $V$ of $...
David E Speyer's user avatar
10 votes
Accepted

Different Bialgebra/Hopf algebra structures on coalgebras

Yes: if $k$ is a field of characteristic 2, let $C$ be the coalgebra over $k$ spanned by 1, $x$, $y$, and $z$ with $x$ and $y$ primitive, $\Delta z = z \otimes 1 + 1 \otimes z + x \otimes y + y \...
John Palmieri's user avatar
10 votes
Accepted

An algebra with more than one Frobenius algebra structure

If $A$ is a Frobenius $K$-algebra and $\lambda\colon A\to K$ is a Frobenius form, then the Frobenius forms are the mappings of the form $a\mapsto \lambda(ua)$ with $u$ a unit of $A$. One way to see ...
Benjamin Steinberg's user avatar
9 votes
Accepted

The dual of a dual in a rigid tensor category

As Tobias said in his answer, a good place to look for examples is in endofunctor categories with composition as the monoidal product, where duals are adjoints. But another way to get a rigid ...
Mike Shulman's user avatar
9 votes

The tensor product of two monoidal categories

If your categories are locally finite abelian, I think you are looking for the Deligne tensor product of $\mathcal{M}$ and $\mathcal{N}$. The Deligne tensor product $\mathcal{M}\boxtimes\mathcal{N}$ ...
Robert McRae's user avatar
9 votes

Is $Tor_A(k,k)$ a bicommutative Hopf algebra?

This is not true. Consider the algebra $A=T(V)/V^{\otimes 2}$, it is a commutative algebra whose augmentation ideal has zero multiplication. We have $\mathrm{Tor}_A(k,k)\cong T(V[1])$ with the shuffle ...
Vladimir Dotsenko's user avatar
9 votes
Accepted

Characterizing discrete quantum groups

Also the implication (2) $\Rightarrow$ (1) holds and can be proven as follows. Denote by $\mathcal{C}$ the category of all finite dimensional, nondegenerate $*$-representations of $M$. The morphisms ...
Stefaan Vaes's user avatar
  • 4,011

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