22
votes
Accepted
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 ...
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&...
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 ...
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 ...
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 $...
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\...
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 ...
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 ...
12
votes
Accepted
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 ...
12
votes
Accepted
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
10
votes
Accepted
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, ...
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}...
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 $...
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 ...
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 ...
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 ...
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 $...
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 \...
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 ...
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 ...
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 $\mathcal{M}\boxtimes\mathcal{N}$ ...
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 ...
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 ...
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