**7**

votes

**4**answers

652 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 ...

**5**

votes

**1**answer

874 views

### What kind of structures allow Galois descent?

EDIT: Question solved.
Let me explain what I mean.
The classical formulation of Galois descent, e. g. in Crawley-Boevey's "Cohomology and central simple algebras", uses the following notion:
...

**3**

votes

**1**answer

258 views

### Cocyles for abelian extensions

Suppose we have an abelian extension of Hopf algebras,
$$k \rightarrow k^G \rightarrow A \rightarrow kF \rightarrow k.$$
According to the general theory there is a left action of $F$ on $G$ and a ...

**0**

votes

**1**answer

424 views

### How is the coradical filtration defined?

I have seen the coradical filtration of a coalgebra $C$ defined as follows:
$C_0 = \text{sum of all simple subcoalgebras of }C$;
for any $n\geq 1$, let $C_n$ be $\Delta^{-1}\left(C\otimes ...

**1**

vote

**1**answer

167 views

### Generators of the Augmentation Ideal (Counit Kernel)

For the Hopf algebra $SL_q(N)$ it is clear that the kernel of the counit contains the ideal generated by the elements $(u^i_i-1)$ and $u^i_j$, for $i \neq j$. However, I cannot seem to arrive at at ...

**3**

votes

**2**answers

233 views

### A Triangular (Non trivial) quasi-Hopf algebra

What would be a good (as "easy" as possible) example of a Triangular (Non trivial) quasi-Hopf algebra? By trivial I mean the quasi strcture not to be trivial, but if the triangular structure is ...

**0**

votes

**1**answer

116 views

### Gradings Induced by Coactions?

A "well-known" fact is that a ${C}_q[U(1)]$-coaction on a vector space $V$ induces a $Z$-grading. I don't see why this is so. Clearly, a ${C}_q[U(1)]$-grading will induce a $Z$-grading. But why does ...

**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

371 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

672 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

457 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

300 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

342 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 ...

**10**

votes

**1**answer

545 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

273 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

502 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

152 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

734 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

419 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

122 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

188 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

469 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 ...

**10**

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

259 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

382 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

151 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

226 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

475 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

236 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

654 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

533 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

645 views

### What is Taft algebra?

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

**4**

votes

**3**answers

715 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

477 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

445 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

475 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

348 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

406 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

355 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

649 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

392 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

250 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

339 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 ...