Questions tagged [hopf-algebras]

A Hopf algebra is a vector space $H$ over a field $k$ endowed with an associative product $\times:H\otimes_k H\to H$ and a coassociative coproduct $\Delta:H\to H\otimes_k H$ which is a morphism of algebras. Unit $1:k\to H$, counit $\epsilon:H\to k$ and antipode $S:H\to H$ are also required. Such a structure exists on the group algebra $k G$ of a finite group $G$.

Filter by
Sorted by
Tagged with
12 votes
3 answers
781 views

Subalgebra of a group algebra

Let $k$ be a field, $G$ a finite group, and $k[G]$ the group algebra. Let $A$ be a subalgebra of $k[G]$. In general, $A$ is not the group algebra of some subgroup $H$ of $G$. Question: Is there any ...
Student's user avatar
  • 5,008
12 votes
2 answers
565 views

On the isomorphism problem of enveloping algebras

Let $\mathfrak{g}$ and $\mathfrak{g}'$ be Lie algebras. It is known that if $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as associative algebras, then it is not necessarily true that $\mathfrak{g}\cong \...
Mathematician 42's user avatar
12 votes
2 answers
1k views

Is there a canonical Hopf structure on the center of a universal enveloping algebra?

Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$. Define $\mathcal Z(\mathfrak g)$ to be the center of the universal enveloping algebra $\mathcal U\mathfrak g$, and define $(\...
Theo Johnson-Freyd's user avatar
12 votes
1 answer
785 views

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

Let $R$ be a regular ring over a field of char 0. Let $X=Spec R$ and $D=\mathcal{D}_X$ the algebra of differential operators over it. The overall vague question is what kind of algebraic object is $...
Saal Hardali's user avatar
  • 7,549
12 votes
1 answer
810 views

Comparing two similar procedures for quantizing a Casimir Lie algebra

My primary reference for this question is the very good book Quantum Groups and Knot Invariants by C. Kassel, M. Rosso, and V. Turaev. I'm also drawing from P. Etingof and O. Schiffmann, Lectures on ...
Theo Johnson-Freyd's user avatar
12 votes
0 answers
181 views

Hopf-Galois extensions where the "extension" is a module?

For $H$ a Hopf-algebra, an $H$-Hopf-Galois extension is a map of rings $\phi\colon\thinspace A\to B$ such that $H$ coacts on $B$ over $A$, $B\otimes_AB\cong B\otimes H$, and the cofixed points, or the ...
Jonathan Beardsley's user avatar
12 votes
0 answers
603 views

Given an algebra, can it be realized as a block of a Hopf algebra?

During a classification problem I came across a set of algebras given as the path algebra of a quiver with relations. As an example the local ones: $k\langle x,y\rangle/x^2,y^2, xy-qyx$, where $q\in k$...
Julian Kuelshammer's user avatar
11 votes
4 answers
4k views

Hopf Algebras and Quantum Groups

I have studied graduate abstract algebra and would like to learn about Hopf algebras and quantum groups. What book or books would you recommend? Are there other subjects that I should learn first ...
11 votes
5 answers
2k views

Structure of the adjoint representation of a (finite) group (Hopf algebra) ?

Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation. Question 1: what is known about this representation ...
Alexander Chervov's user avatar
11 votes
2 answers
1k views

Quantized Enveloping Algebras at $q=1$

As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation $$ [E,F] = \frac{K-K^{-1}}{q-q^{-1}}. $$ To address this problem, one has ...
Antonio Nogueria's user avatar
11 votes
2 answers
2k views

Cartier-Kostant-Milnor-Moore theorem

If $k$ is an algebraically closed field of characteristic zero and $H$ is a cocommutative Hopf algebra, then $$ H \cong U(P(H)) \ltimes kG(H). $$ What happens if the field is not algebraically closed? ...
a213f's user avatar
  • 113
11 votes
2 answers
624 views

A quantum Grothendieck group?

Question 1: Given a co-commutative bialgebra, does there exist a sort of Grothendieck group type construction? Presumably this should take the form of a functor from co-commutative bialgebras to hopf ...
Ollie's user avatar
  • 1,391
11 votes
1 answer
567 views

Hopf algebras and bijective antipodes

By a theorem of Larson and Sweedler, the antipode of every finite-dimensional Hopf algebra is bijective. My question is the following: Is it true that in every noetherian Hopf algebra the antipode ...
warren's user avatar
  • 275
11 votes
1 answer
275 views

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

If $B$ is a subalgebra of $A$, you can ask whether the $B$-module structure on $B$ can be extended to give an $A$-module structure on $B$. W H Lin, in his 1973 PhD thesis at Northwestern, showed that ...
John Palmieri's user avatar
11 votes
3 answers
762 views

Faithfully flat descent over Hopf algebras in terms of comodule structures

Let $A$ be a (finite-dimensional graded cocommutative) Hopf algebra over a field $k$, $E$ be a Hopf subalgebra, and $R=A \otimes_E k$. Then the comultiplication on $A$ induces a coalgebra structure ...
Eric Wofsey's user avatar
  • 30.7k
11 votes
3 answers
315 views

Unbiased Hopf algebras

In category theory, a notion of monoidal category in which every sequence $X_1, \ldots , X_n$ ($n\ge 0$) of objects has a specified product is called an ``unbiased monoidal category'' (see Section 3.1 ...
André Henriques's user avatar
11 votes
2 answers
587 views

Antipode of Hopf algebra in braided monoidal category is an algebra antihomomorphism?

I realize that the question posed in the title has already been addressed here: Identities that connect antipode with multiplication and comultiplication, where the graphical calculus proof provided ...
BarnardIsOnline's user avatar
11 votes
1 answer
552 views

Generators of the Odd Dimensional Quantum Spheres

As is well-known, the $(2N-1)$-quantum sphere $S^{2N-1}_q$ is defined to be the invariant subalgebra of $SU_q(N)$ under the coaction $\Delta_R = (id \otimes \pi) \circ \Delta$, where $\Delta$ is the ...
Janos Erdmann's user avatar
11 votes
1 answer
1k views

When are representation rings special lambda-rings? (variations of an old question)

Status: Questions 2 and 4 answered in the negative. Questions 1 and 3 ARE STILL UNANSWERED, despite previous claims. On the third page of Wolfang K. Seiler's paper "lambda-rings and Adams operations ...
darij grinberg's user avatar
11 votes
1 answer
869 views

Connected graded involutive bialgebras (Hopf algebras) sought (for Dynkin idempotent checking)

Again, there is a general and a concrete question: Concrete question. Let $H$ be a connected graded bialgebra over a commutative ring $k$ with unity (where graded means $\mathbb N$-graded). Then, it ...
darij grinberg's user avatar
11 votes
0 answers
171 views

Natural knot homology

All knot homology theories I've seen share a flaw: their definitions explicitly use some combinatorial choices (such as a diagram presentation). The coin, however, has two sides and the other one is ...
Mikhail Shkolnikov's user avatar
11 votes
0 answers
287 views

Is there a homotopy coherent analogue of Dieudonné modules?

Let $K$ be a perfect field with characteristic $>0$ and $\mathcal{H}$ the category of graded connected abelian hopf algebras over $K$. By a theorem of Schoeller there is a canonical equivalence ...
Hadrian Heine's user avatar
11 votes
0 answers
277 views

Is there a non-Kac complex finite dimensional semisimple Hopf algebra?

A complex (finite-dimensional) Hopf algebra is said to be a Kac algebra if it is a ${\rm C^{\star}}$-algebra in such a way that the comultiplication $\Delta$ is a $\star$-homomorphism. Obviously, a (...
Sebastien Palcoux's user avatar
11 votes
0 answers
660 views

What is the role of fiber functor in Deligne's theorem on Tannakian categories?

The theorem states that, for a field $k$ of characteristic 0, any $k$-linear tensor category with $End(1)=k$ satisfying a condition that each object is annihilated by a Schur functor, is equivalent to ...
Anonymous's user avatar
  • 223
10 votes
3 answers
1k views

Hopf structure on the universal enveloping of a super Lie algebra

The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the ...
Nadia SUSY's user avatar
10 votes
3 answers
1k views

About the classification of commutative and of cocommutative, fin. dim. Hopf algebras

I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f....
Konstantinos Kanakoglou's user avatar
10 votes
1 answer
493 views

Hopf algebra with a non-invertible antipode

What is an example of a Hopf algebra with a non-invertible antipode?
Adam Bondal's user avatar
10 votes
4 answers
1k views

Twist of a group Hopf-algebra

Let $G$ be a finite group with identity element $e$, and $C[G]$ the ring of complex-valued functions on $G$, with pointwise addition and multiplication. Then $C[G]$ is naturally a Hopf algebra, with ...
Marty's user avatar
  • 13.1k
10 votes
1 answer
891 views

The coproduct on the cohomology of a Hopf algebra

If $H$ is a Hopf algebra over a field $k$, the Hopf algebra cohomology $\mathrm{Ext}_H(k,k)$ is —as usual— an algebra, but the Hopfness of $H$ turns it into a Hopf algebra. Is there a reference on ...
Mariano Suárez-Álvarez's user avatar
10 votes
1 answer
1k views

Distributions on product spaces

I hope this is suitable to MO. Question. Let $X$ and $Y$ be two open sets in $\mathbb R^n$ and $\mathbb R^m$, respectively. In what sense can we consider $\mathcal{D}^{\prime} \left(X\times Y\right)$ ...
darij grinberg's user avatar
10 votes
2 answers
876 views

An explicit description of $\operatorname{gr}(k \cdot G)$ for the filtration induced by the augmentation ideal?

Let $A$ be any bialgebra (associative, unital, etc.) over a ring $k$. Then among other things it has a counit $\epsilon : A \to k$, and hence an augmentation ideal $I = \ker \epsilon$, which is a ...
Theo Johnson-Freyd's user avatar
10 votes
1 answer
817 views

Is there a "correct" general setting for the principle: "tensoring any object with a projective object yields another projective"?

Apparently this principle was first formulated for left modules over the group algebra $A=kG$ of a finite group, where $k$ is a field of characteristic $p>0$ dividing $|G|$. (See Exercise 2 on p. ...
Jim Humphreys's user avatar
10 votes
1 answer
837 views

Is there a good reference for the relationship between the Yangian and formal based loop group?

For every finite dimensional semi-simple Lie group $\mathfrak{g}$, we have a loop algebra $\mathfrak{g}[t,t^{-1}]$. This loop algebra has a natural invariant inner product by taking the residue at ...
Ben Webster's user avatar
  • 43.9k
10 votes
1 answer
526 views

Tannakian formalism for topological Hopf algebras

Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of Etingof-...
Sinan Yalin's user avatar
  • 1,589
10 votes
1 answer
515 views

Functoriality of the Hopf dual

Given Hopf $\mathbb{C}$-algebra $H$, it's Hopf dual $H^o$ is the largest Hopf algebra contained in $H^*$, the $\mathbb{C}$-linear dual of $H$. (This is well known to be well-defined, see for example ...
Nadia SUSY's user avatar
10 votes
1 answer
181 views

Exceptional Quantum Groups as FRT-Algebras

Let $\frak{g}$ be a simple Lie algebra of A,B,C,or D series type. Moreover, let $U_q(\frak{g})$ be its Drinfeld-Jimbo quantized enveloping algebra, and $G_q$ the quantized enveloping algebra. As is ...
Alesandro Levi's user avatar
10 votes
0 answers
391 views

Lazard's theorem and Hopf structures on the polynomial algebra

Let $k$ be an algebraically closed field of characteristic $0$. A well-known result of Lazard's states that an algebraic group which is isomorphic as a variety to an affine space is unipotent (M. ...
Paul Gilmartin's user avatar
10 votes
0 answers
283 views

What's the relation between half-twists, star structures and bar involutions on Hopf algebras?

A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in ...
Manuel Bärenz's user avatar
10 votes
0 answers
258 views

Are local finite dimensional Hopf algebras symmetric?

Recall that a finite dimensional algebra $A$ over a field $K$is called a Frobenius algebra in case $A \cong D(A)$ as right modules, where $D(A) \cong Hom_K(A,K)$. In case $A \cong D(A)$ as bimodules, ...
Mare's user avatar
  • 25.8k
10 votes
0 answers
460 views

Bicommutative Hopf algebras have internal hom objects. What are they?

Let $k$ be a field and $\mathcal H$ the category of conilpotent bicommutative Hopf algebras over $k$. (Conilpotent means that every element eventually becomes zero if you apply the reduced ...
Tilman's user avatar
  • 6,032
9 votes
4 answers
2k views

The tensor product of two monoidal categories

Given two monoidal categories $\mathcal{M}$ and $\mathcal{N}$, can one form their tensor product in a canonical way? The motivation I am thinking of is two categories that are representation ...
Nadia SUSY's user avatar
9 votes
4 answers
972 views

The dual of a dual in a rigid tensor category

For a rigid tensor category $\cal{C}$, can it happen that, for some $X \in {\cal C}$, we have that $X$ is not isomorphic to $(X^{*})^*$, for $*$ denoting dual? If so, what is a good example.
Max Schattman's user avatar
9 votes
2 answers
1k views

Algebra in a category

I am try to understand the concept: an algebra in a category. Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. $A$ is an algebra in $\mathcal{C}$ means the multiplication $m: A \...
Jianrong Li's user avatar
  • 6,101
9 votes
3 answers
2k views

Tensor product of linear mappings versus chain complexes

A chain complex of vector spaces $X_k$ is a sequence of linear mappings $\dots \overset{d_{k-1}}{\longrightarrow} X_k \overset{d_{k}}{\longrightarrow} X_{k+1} \overset{d_{k+1}}{\longrightarrow} \dots$...
shuhalo's user avatar
  • 4,736
9 votes
2 answers
905 views

If associated-graded of a filtered bialgebra is Hopf, does it follow that the original bialgebra was Hopf?

Warning: older texts use the word "Hopf algebra" for what's now commonly called "bialgebra", whereas now "Hopf" is an extra condition. So as to avoid any confusion, I'll ...
Theo Johnson-Freyd's user avatar
9 votes
1 answer
329 views

An inner product approach to Hopf algebras

We fix the standard inner products on $\mathbb{C}^n$ and $\mathbb{C}^n\otimes \mathbb{C}^n$. Is there an algebra structure on $\mathbb{C}^n$ with multiplication $m$ such that the adjoint operator $m^...
Ali Taghavi's user avatar
9 votes
2 answers
393 views

Hopf algebra kernels vs. algebra kernels

Let $f: H_1 \rightarrow H_2$ be a map of graded connected cocommutative Hopf algebras over a perfect field. Let $H \subset H_1$ be the Hopf algebra kernel of $f$, and let $I \subset H_1$ be the ...
Nicholas Kuhn's user avatar
9 votes
2 answers
607 views

Definition of subcoalgebra over a commutative ring

Let $k$ be commutative ring and $(C, \Delta)$ be a coalgebra over $k$. Let $D$ be a $k$-submodule of $C$. Notes I'm reading give the following definition: $D$ is called subcoalgebra of $C$ if the ...
user avatar
9 votes
2 answers
431 views

A diagram for understanding action/coaction compatibility in a Yetter-Drinfeld module

For a Hopf algebra $H$ with antipode $S$, let $M$ be a left $H$-module with the action $h \otimes m \mapsto \rho(h,m)$, and also a left $H$-comodule with coaction $\delta \colon m \mapsto m^{(-1)} \...
Mike Pierce's user avatar
  • 1,149
9 votes
1 answer
573 views

In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?

Question: In a closed monoidal abelian category such that the unit object is compact projective, must the tensor product of compact projective objects be compact projective? Recall that an object $X\...
Theo Johnson-Freyd's user avatar

1
2
3 4 5
12