# Questions tagged [bialgebras]

The bialgebras tag has no usage guidance.

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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], ...

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### 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 ...

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### 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\...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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. $r$...

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### 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 ...

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### 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 ...

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### 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^...

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### 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.

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### 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 ...

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### 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. ...

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### 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?

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### 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 $\mathrm{Lin}_k(A,k)$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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}:...

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### 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 ...

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### 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?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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-...

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### 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 ...

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### 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
$$
(b_1b_2) \triangleright a = b_1\triangleright(b_2 \triangleright a).
$$
...

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### 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 ...

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### 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 ...