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 ...

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14
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0answers
141 views

Are there small examples of non-pivotal finite tensor categories?

I'm looking for small concrete examples of non-pivotal finite tensor categories to do some calculations with. Here a finite tensor category is, according to Etingof-Ostrik, a rigid monoidal category ...
12
votes
0answers
485 views

Non-“weakly group theoretical” integral fusion categories?

Can you exclude integral fusion categories of global dimension 210, such that the simple objects have dimensions {1,5,5,5,6,7,7} and the following fusion matrices (I don't write the trivial one) ? ...
12
votes
0answers
488 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 ...
8
votes
0answers
279 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 ...
7
votes
0answers
273 views

Quantum Braid Group

Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting ...
5
votes
0answers
133 views

Is it possible to define Hopf algebra as an object in the category of (associative) algebras?

I think this must be well-known, but I can't find references, so my apologies from the very beginning. Consider first the notion of bialgebra. It is usually defined as an object $B$ in the category ...
5
votes
0answers
170 views

How to compute the abelianization of the representation theory of a Hopf algebra?

I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two. Version 1: Let $(C,\otimes)$ be any monoidal ...
4
votes
0answers
160 views

Existence of a Kac algebra for a given fusion ring in a particular class

A $n$-dimensional Kac algebra (i.e., a Hopf C*-algebra), admits finitely many irreducible representations, whose cardinal $r$ is called its rank, the increasing sequence $(d_{1},d_{2},d_{3}, ..., ...
4
votes
0answers
161 views

Is the “Toeplitz algebra” the representation ring of a Hopf algebra related to SU(2)?

More precisely, does there exist a Hopf algebra $H$ whose category of (finite-dimensional, complex) representations is generated under direct sum and tensor product by two one-dimensional ...
4
votes
0answers
200 views

$U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$

While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is ...
3
votes
0answers
65 views

dual notion to hopf galois extension and properties thereof

Let $H$ be a finite dimensional hopf algebra and $B \subset A$ be an $H$-extension of algebras. We know that the following are equivelant 1) $A \cong B \times_\sigma H$ is a cocycle crossed product ...
3
votes
0answers
62 views

Is multiplication continuous in quantum-SU(2) with respect to the $L^2$-norm

For the Hopf $\ast$-algebra ${\cal O}_q(SU(2)$, with its unique Haar measure $h$, we have an inner product, and a norm, on ${\cal O}_q(SU(2))$ defined by $$ <x,y> : = h(xy^*), ...
3
votes
0answers
96 views

How a unitary corepresentation of a Hopf C*-algebra, deals with the antipode?

Let $\mathcal{A}$ be a Kac algebra (Hopf C*-algebra), with comultiplication $\delta$, counit $\epsilon$ and antipode $S$. The following definition comes from this paper (p51-52) of ...
3
votes
0answers
58 views

Do purification and equivariantization commute?

Suppose that we have an action of a group $G$ on a (quasi-)Hopf algebra $H$, so that we can construct $H\rtimes G$ as in Majid's Cross Products by Braided Groups and Bosonization. It is known that ...
3
votes
0answers
206 views

Quantum Coordinate Algebras at Roots of Unity and Non-Standard Irrep Types

Let $\frak{g}$ be a complex semi-simple Lie algebra of rank $n$, and $U_q(\frak{g})$ the corresponding Drinfeld-Jimbo algebra. As is well-known, for $q$ not a root of unity, the irreducible ...
3
votes
0answers
99 views

Hopf Algebra Pairings and Module-Comodule-Equivalences

Let $\left< , \right> : G \times H \to {\bf C}$ be a dually pairing for two complex Hopf algebras $G$ and $H$. For any (left)-$G$-comodule $(V,\Delta_R)$, we can give $V$ the structure of a left ...
3
votes
0answers
136 views

Is the homotopy of a primitively generated Hopf algebra still primitively generated?

Let $A=\oplus A_n$ be a primitively generated graded Hopf algebra, where each $A_n$ is a simplicial group. This allows us to define the homotopy group $\pi_*(A)$. Question: is the graded Hopf algebra ...
2
votes
0answers
136 views

(Non trivial) coidempotents(Co-$K$-theory)

I was interested to know about coalgebraic version of "Idempotents". So I seached the web and I found the following interesting post : ...
2
votes
0answers
27 views

Coinvariant Complement to Hopf Comodule Morhpism Kernel

Let $(V,\Delta_R)$ be a (right) comodule over a Hopf algebra $H$, and let $f:V \to C$ be a comodule map, where $C$ is viewed as a Hopf algebra in the usual trivial way. Can there exist more that one ...
2
votes
0answers
98 views

quantum deformations of tensor category

I was told that, if I understand correctly, that the enveloping algebra of semisimple Lie algebra admits one family of quantum deformation as Hopf algebra, which was proved by Drinfeld. Anyone can ...
2
votes
0answers
108 views

Is an integral simple fusion ring, categorifiable?

A fusion ring $\mathcal{F}$ (see here p 28) is integral if the Perron-Frobenius dimension $d(h_i)$ of its basic elements $\{h_1,...,h_r\}$, are integers. Its rank is $r$ and its dimension is $\sum ...
2
votes
0answers
83 views

integrals in Hopf algebras

Let $H$ be a Hopf algebra and $g$ be a group-like element in $H^{*}$. Define $L_{g}=\{x\in H\mid hx=g(h)x,~\forall h\in H\}$. If $H$ is not semisimple, when can we find a group-like element $g$ in ...
2
votes
0answers
55 views

2-cocycles/Bigalois-objects over nontrivial liftings

It is easy to extend group-2-cocycles to smash-products with Nichols algebras over the group (just trivially). The same certainly doesn't work for nontrivial liftings. As I would like to check a ...
2
votes
0answers
181 views

dimension of induced comodule

Let $\pi : G --> H$ be epimorphism of Hopf superalgebras, where $G$ be an quantum super group of function on $GL(m|n)$, $H$ be an quantum group of function on $GL(m) \otimes GL(n)$; $W$ an finite ...
2
votes
0answers
254 views

What happens geometrically when you take associated-graded (or complete, …) of a group ring at its augmentation ideal?

I am interested in the following functor from Monoids (in $\text{Set}$) to Graded Lie Algebras (over a fixed field of characteristic $0$). (By "graded" I mean only that my Lie algebras have some ...
1
vote
0answers
41 views

For Finite Dual when is $(A \otimes A)^o = A^o \otimes A^0$?

Let $A$ be any $k$-algebra. The finite dual or restricted dual of $A$ is $$ A^o = \{f \in A^* ~ | ~ f(I)= 0, \text{ for some ideal } I \subseteq A, \text{ such that } \text{dim}_k(A/I) < \infty\}. ...
1
vote
0answers
40 views

Alternative Generating Sets of the Quantum Special Linear Group

The Hopf algebra ${\cal O}_q(SL(N))$ has as generators the elements $u^i_j$, subject to certain $q$-relations. Moreover, the antipode is bijective, with square equal to a multiple of the identity on ...
1
vote
0answers
82 views

Action of Landweber-Novikov algebra on infinite polynomial ring

Denote $\text{Aut}(\hat{{\mathbb A}}^1)$ be the affine group over ${\mathbb Z}$ that sends some ring $R$ to the strict automorphisms of $R[[t]]$, i.e. those of the form $X\mapsto X + r_1 X^2 + r_2 X^3 ...
1
vote
0answers
77 views

faithful modules of algebraic group

Let $G$ be a linear algebraic group over a field $k$. $k[G]$ is the coordinate ring of $G$. $k[G]^{*}$ is the dual algebra of the coalgebra $k[G]$. $H=k[G]^{\circ}$ is the finite dual of the Hopf ...
1
vote
0answers
94 views

How simplify the pentagonal equation from two fusion rings?

A semi-simple finite dimensional Hopf algebra $\mathbb{A}$, and its dual $\mathbb{A}^{*}$ produce two fusion categories $\mathcal{C}$ and $\mathcal{C}^{*}$ and then two fusion rings $\mathcal{R}_{1}$ ...
1
vote
0answers
155 views

universal enveloping algebras and commutator subalgebras

Let $A$ and $B$ are Lie subalgebras of a Lie algebra $L$. $U(A)$, $U(B)$ and $U(L)$ are the universal enveloping algebras of $A$, $B$ and $L$, respectively. Let $[A, B]$ be the Lie subalgebras ...
1
vote
0answers
140 views

Classification of Hopf algebra with exactly two 1-dimensional modules

Is there a classification of indecomposable non-semisimple finite dimensional Hopf algebras with exactly two 1 dimensional modules? If not, is there one when all simple modules are 1 dimensional and ...
1
vote
0answers
83 views

Equivalence between Algebraic Semi-group Structures and Coalgebra Structures for an Algebraic Variety?

I was looking at this old question Hopf algebra and group structure correspondence for algebraic varieties which says that there exists an equivalence between algebraic group structures on an ...
1
vote
0answers
109 views

Hopf Comodule Decompositions and Coinvariant Elements

For $H$ an Hopf algebra, and $V$ a right $H$-comodule with coaction $\Delta_R$, for which there exists a vector space decomposition $V=V_+ \oplus V_-$. Are the following two statements equivalent? ...
1
vote
0answers
95 views

Group-like Elements in a Coquasitriangular Bialgebra

What do we know about group-like elements in coquasitriangular bialgebras? In particular, we know that a commutative bialgebra in which all group-like elements are invertible is a Hopf algebra (see ...
1
vote
0answers
128 views

There is nonzero primitive element in finite dimensional pointed hopf algebra over C???

I'm huge confused! There is nonzero primitive element in finite dimensional pointed hopf algebra over complex field??? I find in several articles,it is said that a is primitive,so a=0. I will ...
1
vote
0answers
247 views

Coideals of Hopf algebra coming from right (left) coideals K->K^+

If $H$ is a Hopf algebra over a field and $K$ is a a right or left coideal of $H$ then $K^+=K\cap\ker\epsilon$ is a coideal of $H$. Does this hold when $k$ is a commutative ring? If not what is a ...
1
vote
0answers
326 views

Complement to the Kernel of a Hopf Algebra Map

Given two Hopf algebras $A,B$ over a field $k$, and a Hopf algebra map $\pi:A \to B$, are there any general tricks for finding a complement to the kernel of $\pi$. That is, how can one find a subspace ...
1
vote
0answers
129 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 ...
1
vote
0answers
334 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 ...
0
votes
0answers
101 views

Flatness over Hopf subalgebra

Let $A\subset B$ be flat Hopf algebras over a Dedekind ring $R$. J. Moore has proved in the article Compléments sur les algèbres de Hopf, John C. Moore, Séminaire Henri Cartan (1959-1960), Volume: ...
0
votes
0answers
167 views

polynomial representation of $sl_{2}(k)$

Let $k$ be an algebraic closed field of characteristic 0. We write $$X=\left( \begin{array}{ccc} 0 & 1\\ 0 & 0\\ \end{array} \right),~~ Y=\left( \begin{array}{ccc} 0 & 0\\ 1 & 0\\ ...
0
votes
0answers
69 views

Degree of a commutator in a hyperalgebra or enveloping algebra

Consider a semisimple algebraic group $G$ over an algebraically closed field of arbitrary characteristic and let $\bar U(G)$ denote its hyperalgebra (ie, the restricted Hopf dual of the coordinate ...
0
votes
0answers
199 views

Why are there two Hopf algebra structures on a Kac--Moody Algebra.

For a Lie algebra $\frak{g}$, we will recall that its universal enveloping algebra $U(\frak{g})$ has a Hopf algebra structure given by $$ \Delta(x) := 1 \otimes x + x \otimes 1, ~~~ \epsilon(x) = 0, ...
0
votes
0answers
67 views

When is a Surjective Comodule Endomorphism an Automorphism?

Given a Hopf algebra $H$, a left $H$-comodule $V$, and a surjective comodule endomorphism $f: V \to V$. Can somebody give: (i) a set of neccessary, or sufficient, or both neccessary and sufficient, ...
0
votes
0answers
227 views

Annulator of Tensor Power in a Quantum Group

There is a little question haunted me for few days. I will be grateful to anyone who can give me any clue how to solve it. Let $V$ be a nontrivial module of $\mathrm{U}_q(\mathfrak{g})$ (the ...
0
votes
0answers
240 views

Pure submodules

Is the dual of a pure module also pure? Suppose $H$ is a Hopf algebra over a field $K$ and $S$ is a ring which has a right $H$-action. If ${}_HA$ is a pure left $H$-submodule, will $S\otimes_H A$ be ...