**2**

votes

**1**answer

221 views

### Hopf Algebras/Rings, A Question of Terminology

I'm reading $\textit{The Hopf Ring for Complex Cobordism}$ by Ravenel and Wilson where they discuss the notion of group and ring objects over a category. They say that a hopf algebra over a ring in ...

**3**

votes

**1**answer

336 views

### Does There Exists a General Quantum Casimir Extending the $U_q({\mathfrak sl}_2$ Case?

As is well known (see Kassel), when $q$ is not a root of unity, the centre or the quantum enveloping algebra $U_q({\mathfrak sl}_2)$ of ${\mathfrak sl}_2$ is generated by the element
$$
C_q = EF + ...

**15**

votes

**1**answer

657 views

### The Major Families of Quantum Groups

If we define a quantum group to be a quasi-triangular or coquasi-triangular Hopf algebra, then what are the major families of quantum groups?
Of couse, to start with we have the h-adic completions ...

**3**

votes

**1**answer

454 views

### Prove that $U^*U=UU^*=1$ for $U_q(N,C)$

Let $u^i_j$, $i,j = 1, . . . N$, and det$_q^{-1}$ be the standard generators of the quantum group $U_q(N,C)$, and define the matrices $U$ and $U^{\ast}$ by setting $U_{ij} := u^i_j$ and ...

**2**

votes

**1**answer

245 views

### Hopf algebra and group structure correspondence for algebraic varieties

Let $V$ be a real algebraic variety and let ${\cal O}(V)$ denote its algebra of regular functions. If we put a group structure on $V$ (not necessarily an algebraic group structure) it will induce a ...

**4**

votes

**2**answers

298 views

### Sub-Hopf algebras of group algebras

Let $k$ be a field and $G$ a finite group. Is every sub-Hopf algebra over $k$ of the group algebra $k[G]$ of the form $k[U]$ for a subgroup $U$ of $G$ ?

**2**

votes

**2**answers

341 views

### The relation $S(u^1_i)u^j_1 = q^{-1}u^j_1S(u^1_i)$

In the Hopf algebra $SL_q(N)$, it can be shown, using direct calculations, that $S(u^1_i)u^j_1 = q^{-1}u^j_1S(u^1_i)$. Can anyone see a more elegant way of establishing this?
Moreover, does anyone ...

**9**

votes

**1**answer

510 views

### Hopf algebras as cohomology of $\mathbb{CP}^\infty$, $\Omega S^3$ and related $H$-spaces

Let me begin by a couple of questions :
Consider a graded abelian group $V=\oplus_{i\geq 0} V_i$ such that $V_{2i}=\mathbb{Z}$ and $V_{\textrm{odd}}=0$. What are the possible Hopf algebra ...

**5**

votes

**1**answer

268 views

### Do dualizable Hopf algebras in braided categories have invertible antipodes?

A classical result of Larson and Sweedler says that a finite dimensional Hopf algebra over a field has invertible antipode. Does this result extend to the setting of Hopf algebras in braided ...

**1**

vote

**1**answer

1k views

### A question about the smash product

Let $K$ be an algebraically closed field and $H$ a finite dimensional semisimple-cosemisimple Hopf $K$-algebra, and let $A$ be a finite dimensional $H$-module algebra and $H^{\ast}= Hom_{K}(H, K)$.
...

**10**

votes

**1**answer

497 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. ...

**1**

vote

**0**answers

149 views

### Duflot-type theorem for Hopf algebras ?

In group cohomology Duflot's theorem states that the depth of the mod p cohomology ring of a finite group is greater than or equal to the p-rank of the center of a Sylow p-subgroup.
Is there a ...

**3**

votes

**3**answers

721 views

### Innocent question on tensor products of modular representations

Let $K$ be a field (of course, of positive characteristic, unless you want a trivial question). Let $G$ be a finite group, and $V$ and $W$ be two completely reducible (finite-dimensional) ...

**1**

vote

**1**answer

416 views

### The Killing Form for Co-Quasi-Triangular Hopf Algebras

For a co-quasi-triangular Hopf algebra $H$, with universal $r$-form $r$, there exists an important map $Q$ defined by
$$
Q:H \otimes H \to k, ~~~~~~h \otimes g \mapsto r(g_{(1)}\otimes ...

**0**

votes

**1**answer

120 views

### Action of Co-quasi-triangular Universal r-form on $a \otimes 1$

A very easy question I can't seem to answer: For a universal r-form on a co-quasi-triangular Hopf algebra why is $r(a \otimes 1) = r(1 \otimes a) = \epsilon(a)$?

**1**

vote

**1**answer

185 views

### Establishing the Co-Quasi- Triangular Structure of FRT Algebras

Let $V$ be a finite dimensional vector space and let $R$ be a linear invertible mapping from $V \otimes V$ to itself. If we fix a basis $\{e_i\}_{i=1,2, \cdots ,n}$ then we have $N^4$ complex numbers ...

**0**

votes

**5**answers

468 views

### Map constructed from the coquasitriangular structure of SLq(2) which appears not to respect the standard commutation relations

Let $A$ be a Hopf algebra dually paired with a quasi-triangular Hopf algebra $B$. If $x$ is some fixed element of $A$, then we can define a linear map
$$
P_x: A \to \mathbb{C}
$$
by setting
$$
P_x:a ...

**17**

votes

**2**answers

1k views

### What is the Hopf algebra structures in the homology of the based loop spaces of $E_7$ and $E_8$?

Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free ...

**9**

votes

**2**answers

2k views

### Hopf algebra duality and algebraic groups

Background:
Let $G$ be a linear algebraic group over an algebraically closed field $k$ and let $I \subseteq k[G]$ be the ideal of the identity element. The hyperalgebra $U(G)$ of $G$ is defined to be ...

**3**

votes

**3**answers

1k views

### When are the homology and cohomology Hopf algebras of topological groups equal?

Suppose we have a topological group $G$, then the multiplication map $\mu$ and the diagonal map $\Delta$ provide the cohomology $H^\ast(G;R)$ (with Pontryagin coproduct and cup coproduct) and homology ...

**2**

votes

**1**answer

253 views

### Semisimple Hopf algebras with commutative character ring

Suppose that $A$ is a semisimple Hopf algebra with a commutative character ring. Does it follow that $A$ is quasitriangular, i.e $\mathrm{Rep}(A)$ is a braided tensor category?
I think I 've seen ...

**1**

vote

**0**answers

374 views

### Ring objects in the category of cocommutative coalgebras (aka Hopf rings).

I have recently been doing some calculations in topology which are naturally expressed in terms commutative ring objects in the category of cocommutative coalgebras. These have been studied for quite ...

**2**

votes

**1**answer

147 views

### Formula for the Matrix Elements of the Inverse of special linear Universal R-Matrix of Uq(sln)

Motivated by this question, I'm going to ask a much more direct one: Let $R$ be a universal R-matrix for the quasitriangular Hopf algebra ${\cal U}_q ({\mathfrak sl}_N)$, $~$ $R ^{-1}$ its inverse, ...

**2**

votes

**1**answer

222 views

### Reference for the Hecke relation for the universal R-matrix

I've come across a reference in a paper to the
Hecke relation for the universal R-matrix of a quasi-triangular Hopf algebra.
I've looked around, standard references, online etc, but can't seem ...

**10**

votes

**1**answer

467 views

### The Inverse of a Universal R-Matrix for Quantized Universal Enveloping Algebra of sl2 and the Dual Pairing with SUq(2)

I have recently begun to study quasi-triangular structures and have come across a problem I can't resolve. Let ${\cal U}_q({\mathfrak sl}_N)$ denote the quantised enveloping algebra of ${\mathfrak ...

**8**

votes

**4**answers

2k views

### What is the universal enveloping algebra?

Let ${\mathfrak g}$ be a Lie algebra in a symmetric monoidal category enriched over $K$-vector spaces, i.e., in particular, hom-s are $K$-vector spaces (where $K$ is a field of characteristic zero). ...

**1**

vote

**2**answers

231 views

### Group and Hopf Algebra Structures for Projective Varieties

Let $V$ be a projective (or affine) variety. Does there exist a bijective correspondence between group structures on $V$ and Hopf algebra structures on the coordinate ring of $V$?

**17**

votes

**4**answers

1k views

### Does the forgetful functor {Hopf Algebras}→{Algebras} have a right adjoint?

(Alternate title: Find the Adjoint: Hopf Algebra edition)
I was chatting with Jonah about his question Hopf algebra structure on $\prod_n A^{\otimes n}$ for an algebra $A$. It's very closely related ...

**6**

votes

**2**answers

652 views

### Hopf algebra structure on $\prod_n A^{\otimes n}$ for an algebra $A$

For a finite dimensional $k$-algebra $A$, each $A^{\otimes n}, n \geq 0$ is a $k$-algebra ($A^0 = k $). Let $T= \prod_{n \geq 0} A^{\otimes n}$. This is a $k$-alegbra with unit $(1,1,\dots)$ and ...

**7**

votes

**1**answer

523 views

### Are groups in (Var/k, rational maps) necessarily algebraic groups?

Let $RVar$ be the category which has complex algebraic verieties as objects and rational maps as morphisms [Edit: for $RVar$ to be a category, the rational maps have to be dominant].
Let's consider a ...

**0**

votes

**1**answer

629 views

### What is Taft algebra?

What is a Taft algebra? Is there any references about the original conception?

**4**

votes

**3**answers

706 views

### Is a bialgebra pairing of Hopf algebras automatically a Hopf pairing?

The following question came up in the course on Quantum Groups here at UC Berkeley. (If you care, I have been TeXing uneditted lecture notes.)
Let $X,Y$ be (infinite-dimensional) Hopf algebras over ...

**6**

votes

**1**answer

470 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 ...

**5**

votes

**5**answers

433 views

### What are the correct axioms for a “weakly associative monoidal functor”?

Definitions and the main question
Recall that a category $\mathcal C$ is monoidal if it is equipped with the following data (two functors, three natural transformations, and some properties):
a ...

**5**

votes

**1**answer

470 views

### Is there a relative version of Tannakian reconstruction?

According to some form of Tannakian reconstruction, given a finite tensor category with a fiber functor to the category of vector spaces, one determines a Hopf algebra by considering tensor ...

**3**

votes

**3**answers

345 views

### Group-Adjoint and Hopf-Algebra-Adjoint Maps

I've reading some introductory quantum group material and am trying to understand the algebra-space correspondence in the classical case. One object I'm stuck on is the adjoint coaction
$$
Ad_R: a ...

**7**

votes

**1**answer

399 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 ...

**3**

votes

**1**answer

224 views

### In what way do exact sequences of Lie ideals integrate to the category of groups?

Please excuse, very naive question:
Suppose $g$ is a topological Lie algebra over Q and $G$ = $exp(g)$ the associated group
(take free group on formal symbols $exp(X)$, X $\in$ $G$, and impose all ...

**2**

votes

**2**answers

353 views

### Finding the Universal Ideal of a (Covariant) Differential Calculus

Let $(\Omega,d)$ be a differential calculus over an algebra $A$. It is easy to show that $\Omega$ is always equal to a quotient of $\Omega_u(A)$, the universal calculus over $A$, by some ideal $N$ of ...

**10**

votes

**2**answers

641 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 ...

**5**

votes

**4**answers

390 views

### Is every monomorphism of commutative Hopf algebras (over a field) injective?

Is it true that any monomorphism of commutative Hopf algebras over a field is injective? Moreover, is it true that any epimorphism of commutative Hopf algebras over a field is surjective?

**2**

votes

**1**answer

246 views

### When does a certain natural construction on monoidal categories yield a Hopf algebra?

Let $\mathcal C = (\mathcal C_0,\mathcal C_1)$ be a (small) strict monoidal category. Pick a field $\mathbb K$, and let $\mathbb K[\mathcal C_1]$ be the vector space with basis the morphism of ...

**6**

votes

**1**answer

333 views

### Transmutation versus operads

A while ago, I was reading Majid's book Foundations of quantum group theory, and Section 9.4 has a rather fascinating description of a Tannaka-Krein reconstruction result for quantum groups. In ...

**4**

votes

**4**answers

942 views

### Apocryphal Maschke theorem?

This may be totally trivial or wrong. I am just posting this because I am sick and tired of trying to understand this myself and I am sure someone out here can just answer it out of his head in 2 ...

**6**

votes

**2**answers

391 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 give my definitions before ...

**7**

votes

**3**answers

2k views

### Faithful characters of finite groups

Related to a previous question I am asking furthermore a proof
for the following:
Question 1: If $\chi$ is a faithful irreducible character of a
finite group $G$ then the regular character of $G$ is ...

**25**

votes

**11**answers

3k views

### What does the generating function $x/(1 - e^{-x})$ count?

Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series
$$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty ...

**22**

votes

**6**answers

2k views

### Why Drinfel'd-Jimbo-type Quantum Groups?

Hopf algebras are pretty easy to motivate, as a not-necessarily-commutative generalization of the ring of functions on an algebraic group (and there are many other ways in which they come up). I like ...

**12**

votes

**6**answers

1k views

### What structure on a monoidal category would make its 2-category of module categories monoidal and braided?

So, many of us know the answer to "what kind of structure on an algebra would make its category of representations braided monoidal": your algebra should be a quasi-triangular Hopf algebra (maybe if ...

**7**

votes

**3**answers

459 views

### Are supervector spaces the representations of a Hopf algebra?

Supervector spaces look a lot like the category of representations of $\mathbb{Z}/2\mathbb{Z}$ - the even part corresponds to the copies of the trivial representation and the odd part corresponds to ...