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Questions tagged [bialgebras]

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Equivalence between bialgebras and finite ring categories with fibre functor

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Rec{Rec}\DeclareMathOperator\End{End}\DeclareMathOperator\Mod{Mod}\newcommand\fdMod[1]{#1\text-{\Mod}^\text{fd}}$In the celebrated [EGNO, Thm 5.2.3], ...
Minkowski's user avatar
  • 571
2 votes
0 answers
141 views

The "big bracket" in Lie bialgebras

I am looking for a well-written document such as a survey article or textbook that explores the subject of the "big bracket". This concept is briefly introduced in the appendix of Yvette ...
user56980's user avatar
  • 442
3 votes
0 answers
513 views

Particular Lie bialgebra structure

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra, and let $r\in\bigwedge^2\mathfrak{g}$ be a solution of the Yang-Baxter equation. The Yang–Baxter equation states that for all $x\in\...
user56980's user avatar
  • 442
2 votes
0 answers
86 views

Quantum groups as bialgebra cohomology classes

My question below is about how to view the quantum group $U_q(\mathfrak{g})$ as a bialgebra cohomology class. Background: If $A$ is a bialgebra, Gerstenhaber and Schack in Bialgebra cohomology, ...
Pulcinella's user avatar
  • 5,565
5 votes
2 answers
231 views

An algebra map between Hopf algebras that does not commute with the counit

Let $(H,\Delta,\epsilon,S)$ be a Hopf algebra. Can there exist an algebra map $\phi:H \to H$ such that $$ \epsilon(\phi(g)) \neq \epsilon(g), ~~~~~ \textrm{ for some } g \in H? $$ Does the anti-pode ...
Lorenzo Del Vecchiopontopolos's user avatar
4 votes
0 answers
191 views

Is it possible to compute Lie bialgebra structures with SageMath?

Is it possible to use SageMath (or some Linux open source program) to compute the bialgebra structures on a given finite dimensional Lie algebra? I wonder if such program can compute all the ...
amine's user avatar
  • 503
3 votes
0 answers
144 views

Is there a classical version of Yetter-Drinfeld modules?

One motivation for the notion of the Drinfeld double $D(H)$ of an Hopf algebra $H$ is that it is defined exactly so that modules over $D(H)$ correspond to Yetter-Drinfeld modules over $H$. If we think ...
Antoine Labelle's user avatar
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0 answers
69 views

Why is $\Pi_r^L$ a non-degenerate Poisson structure on $G\ $?

Let $r \in \bigwedge^2 \mathfrak {g}$ be a skew-symmetric solution of the CYBE (classical Yang-Baxter equation) so that it gives rise to a non-degenrate triangular structure on $\mathfrak {g}$ i.e. $r$...
Anil Bagchi.'s user avatar
5 votes
1 answer
126 views

Bicrossed and bismash product of Hopf algebras

I have been recently studying different methods to construct Hopf algebras. In Theorem IX.2.3 of "Quantum groups" by Kassel the bicrossed product of a pair of matched bialgebras (or Hopf ...
dm82424's user avatar
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13 votes
1 answer
445 views

Hopf algebras vs. Kac algebras

I recently came across Kac algebra. They are roughly Hopf algebras and $C^*$-algebras with compatible structures. It follows from Artin–Wedderburn theorem that every semisimple complex Hopf algebra ...
dm82424's user avatar
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3 votes
0 answers
108 views

A certain Lie algebra associated to a bialgebra

Let $A$ be a bialgebra over a field $k$. The space $A^* = \operatorname{Hom}(A, k)$ possesses an algebra structure, given by the convolution product $f*g = f \otimes g \circ \Delta$. Let $\gamma \in A^...
Jo Mo's user avatar
  • 338
7 votes
2 answers
337 views

Different Bialgebra/Hopf algebra structures on coalgebras

Given a coalgebra $C$, can there exist more than one algebra structure on $C$ giving it the structure of a bialgebra? I will also ask the same question for Hopf algebras.
Alain Rochefort's user avatar
1 vote
0 answers
60 views

Indecomposable comodules

For a Hopf algebra $A$, we say that a comodule $V$ is indecomposable if it is not equivalent to a direct sum of irreducible comodules. $\bullet$ What is an example of a finite dimensional ...
johhnyelgerton's user avatar
1 vote
1 answer
619 views

What is a coalgebra?

A coalgebra is a triple $(A,\Delta,\epsilon)$ consisting of a vector space, a coproduct, and a counit. Now as we all know, just like the unit in an algebra, the counit of a coalgebra is unique, i.e. ...
johhnyelgerton's user avatar
2 votes
1 answer
224 views

Bialgebra maps and Hopf algebra maps

Let $H$ and $H'$ be two Hopf algebras, and let $\phi:H \to H'$ be an bialgebra map. Then is $\phi$ automatically a Hopf algebra map?
johhnyelgerton's user avatar
2 votes
0 answers
110 views

Extending elements of the dual of a Hopf subalgebra to the Hopf dual of the whole algebra

I have a question about Hopf duals. To begin we can remind the definition: For an Hopf algebra $A$ over a field $k$, the Hopf dual $A^{\circ}$ is the subspace of the linear dual $\mathrm{Lin}_k(A,k)$ ...
Alain Rochefort's user avatar
4 votes
1 answer
555 views

Name for a Hopf algebra whose only grouplike element is the identity?

For a $k$-Hopf algebra $H$ and element $h \in H$ is called grouplike is $\Delta(h) = h \otimes h$ and $\epsilon(h)=1_k$ ($\epsilon$ is the counit). The identity $1_H$ is clearly grouplike, but in ...
Spyros Olympopolous's user avatar
3 votes
0 answers
98 views

Fiber product of group rings

Let $K$ be a field. For a group $G$ we write $K[G]$ for the group ring of $G$. Given group homomorphisms $F \to G, H \to G$ is the canonical map $$ K[F \times_G H] \to K[F] \times_{K[G]} K[H] $$ an ...
Hadrian Heine's user avatar
4 votes
0 answers
143 views

Bialgebras from mixed Bruhat sheaves

Let $A = \bigoplus_{i=0}^{+\infty} A_i$ be a graded bialgebra in a braided monoidal category $\mathcal V$. Then, according to Kapranov–Schechtman's article "Parabolic induction and perverse ...
Nicolas Hemelsoet's user avatar
3 votes
1 answer
168 views

Show that a certain element is a linear combination of tensors

I posted this question on MSE but got no answer even after putting a bounty on it, so I figured I can try to ask here. Let $(A, \Delta: A \to A \otimes A)$ be bialgebra (unital and counital) such that ...
user avatar
4 votes
2 answers
190 views

Bialgebras with rigid representation theory

Repost from math.SE since no answer after two months, but feel free to close if not appropriate: Everything is finite-dimensional over a field $k$. Let $B$ be a bialgebra with $B\text{-mod}$ its ...
Jo Mo's user avatar
  • 338
2 votes
1 answer
120 views

Definition of multiplier bialgebra

Consider the following fragments from "An invitation to quantum groups and duality" by Timmerman: Question: In remark 2.1.6 (ii), it is stated that the homomorphism $\Delta\otimes \text{id}:...
user avatar
2 votes
2 answers
405 views

Hopf algebra with a non-grouplike invertible element

What is an example of a Hopf algebra $(H,\Delta,\epsilon)$ containing an invertible element $h$ which is not grouplike: An element $h \in H$ such that $$ \Delta(h) \neq h \otimes h\qquad\text{(not ...
Jake Wetlock's user avatar
  • 1,154
1 vote
1 answer
197 views

Example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra

It's all in the question! What is an example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra?
Jake Wetlock's user avatar
  • 1,154
5 votes
1 answer
710 views

Further developments of Cartier–Gabriel–Kostant–Milnor–Moore Structure Theorem for cocommutative Hopf algebras

A very well-known theorem in Hopf algebra theory (see, for example, Lorenz - A tour of representation theory or the EGNO book (Etingof, Gelaki, Nikshych, and Ostrik - Tensor categories)) states that ...
Ender Wiggins's user avatar
1 vote
1 answer
105 views

On reflexive bialgebras

Let $A$ be a bialgebra. We can consider $A$ as a relfexive algebra (i.e. $A\cong A^{o*}$) or relfexive coalgebra (i.e. $A\cong A^{*o}$ where in each case $o$ denotes what is sometimes called ...
Christoph Mark's user avatar
3 votes
1 answer
429 views

Rigidity for the category of comodules over a Hopf algebra

On this page https://ncatlab.org/nlab/show/rigid+monoidal+category there is a discussion of rigidity (left-right duality) for the catagory of modules over a Hopf algebra. What happens if we look at ...
Max Schattman's user avatar
5 votes
1 answer
578 views

Comultiplication on objects in an (abelian?) category

Looking for example at $R$-modules for some commutative $R$, we have the direct sum and the tensor product acting analogously to addition and multiplication. After studying a little bit about co-...
Adi Ostrov's user avatar
3 votes
1 answer
211 views

turning left modules into right modules over bialgebroids

Let $\mathcal{B}$ be a left R-bialgebroid as defined in https://arxiv.org/pdf/1403.3597.pdf on page 87. Let $M$ be a left $\mathcal{B}$-module. Can $M$ be made a right $\mathcal{B}$-module (as for the ...
1985's user avatar
  • 31
8 votes
1 answer
286 views

Name for the action of a bialgebra on an algebra

Give an algebra $A$, a bialgebra $B$, and an action, that is, a bilinear map $\triangleright: B \times A \to A$ such that $$ (b_1b_2) \triangleright a = b_1\triangleright(b_2 \triangleright a). $$ ...
Tomasz Köner's user avatar
2 votes
1 answer
85 views

does finite dimensional representations of bialgebras separate elements?

let $B$ be a bialgebra over a field (i.e. associative, coassociative, unitary and counitary, maybe it has an antipode or maybe not). If $b\in B$ acts by zero on every finite dimensional ...
Marco Farinati's user avatar
5 votes
1 answer
348 views

Meaning of coinvariants of a comodule

Let $M$ be a comodule over the bialgebra $B$, with structure map $\rho:M \to M \otimes B$. The space of coinvariants is defined as $M^{coB}:=\{m \in M~|~\rho(m)=m\otimes 1_B\}$. A book I'm reading ...
AaronS's user avatar
  • 257