For questions about Hopf algebras and related concepts, such as quantum groups.

learn more… | top users | synonyms

1
vote
0answers
20 views

References about an action $U_q(n) \times \mathbb{C}_q[N] \to \mathbb{C}_q[N]$.

Let $N$ be the unipotent subgroup of a Lie group $G$ consisting of all upper triangular matrices and $n$ the Lie algebra of $N$. I found in some paper that there is an action $U_q(n) \times ...
2
votes
0answers
40 views

A question about the dual map of $A \otimes B \to B$.

Let $A$, $B$ be two algebras. Suppose that we have an action $\varphi: A \otimes B \to B$ and there is a pairing $\psi: A \otimes B \to \mathbb{C}$. The action $\varphi$ induces a map $\delta: B \to ...
2
votes
0answers
22 views

pontrjagin ring of the homology of iterated loop suspension

In The homology of C n+1 spaces, n>=0, F. Cohen, proof of Theorem 3.1 and proof of Theorem 3.2 (p. 228 - 243) I totally do not understand the proofs of these two theorems from page 228 to page 243 ...
0
votes
0answers
24 views

Intrinsic action on dual basis of a quantum group module

TL;DR. Is there a way to describe the action of a quantum group (that is, the action of the $E$'s and $F$'s) on the linear dual of a module (with action given by the antipode) explicitly in terms of a ...
2
votes
0answers
11 views

Hopf algebra associated to $GL_V$ for $V$ infinite rank?

Let $V$ be a free $\mathbb{Z}$-module of rank $n$, and let $F$ be the functor associating to a ring $R$ the group $Aut_R(V \otimes R)$. If $V$ was finite, say $Z^n$, this functor is representable by ...
2
votes
0answers
25 views

Equivalence between modules and comodules

Let $C$ be a coalgebra over a commutative ring $R$, if $C$ is cauchy (f.g. and projective), then there is an equivalence of categories between $\operatorname{Comod}(C)$ and the category ...
6
votes
1answer
49 views

Non-computational proof that $det(A)$ is a unit in $R$ implies $A$ is a unit, for $A \in M_n(R)$.

Quoting from Waterhouse's Introduction to Affine Group Schemes: "...suppose we have a representable functor $G$ [from $k$-algebras to groups], and a map $\Delta: A \to A \otimes A$ giving a ...
1
vote
1answer
35 views

what means 'the realization of a topological category'

In the paper Homology Fibrations and the "Group-Completion" Theorem. page 280 bottom line 10-bottom line 12, what means 'the realization of a topological category'?
0
votes
0answers
13 views

Indecomposables generate the algebra

The following result seems intuitive, but I'm having hard time proving it, or determining if it is, in fact, false. Let $(A, \mu, \eta, \epsilon)$ be an augmented algebra, with augmentation ideal ...
1
vote
0answers
20 views

Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$? In the case of $G = SL_2$, we have $\mathbb{C}[SL_2] = \langle a,b,c,d\rangle / (ad-bc-1 )$ and ...
0
votes
1answer
20 views

A computation for Manin triple.

I am reading the book. I have some questions about the computations in (4.1) on page 40. The computation are in the following. I don't know why $$ ([[e_r^*, e_k], e_s^*]+[e_r^*, ...
2
votes
0answers
23 views

graded Hopf algebra and its dual

I am learning Hopf algebras, and there are two questions as follows: Is the tensor product of two Hopf algebras still a Hopf algebra? Let $A$ be an infinite dimensional algebra. Is the dual ...
0
votes
0answers
9 views

The Hopf algebra of symmetric functions and quasi-symmetric functions

The algebras Sym of symmetric functions and Qsym of quasi-symmetric functions are Hopf algebras. As we know, Sym and Qsym are subalgebra of the form power series ring Q[[x_1,x_2,...]]. Are they the ...
0
votes
0answers
32 views

The maps $\delta: V \to B \otimes V$ and $\delta': V \otimes V^* \to B$.

Let $B$ be a coalgebra and $V$ a vector space. Suppose that we have a coaction $\delta: V \to B \otimes V$. Is the map $\delta$ equivalent to a map $\delta': V \otimes V^* \to B$? Thank you very much. ...
2
votes
0answers
39 views

Reference for Hopf algebra applications to Feynman diagrams

I need to give a talk about Hopf algebras and I would like to give a (at least) 5 minutes introduction using Feynman diagrams as a motivation. I'm looking for a not-so-heavy reference explaining how ...
4
votes
1answer
108 views

An R-matrix in a quasitriangular Hopf algebra

I am new to the theory of Hopf algebra. So I am sorry if the following question has really trivial answer. Suppose that we have a quasi triangular Hopf algebra with the universal R-matrix $R$. It ...
2
votes
1answer
33 views

Motivation behind Quasitriangular Hopf algebra

I would like to know why it is interesting to define the quasi-triangular structure on a Hop algebra. I understand that the pseudo-co-commutative (the existence of an intertwining operator between the ...
1
vote
0answers
20 views

A triangular Hopf algebra and its unitary R-matrix

Why is the R-matrix of a Hopf algebra called unitary when it satisfies the relation $$R^{-1}=R_{12},$$ I would say invertible, why then call it unitary? Is that a nomenclature that maybe comes from ...
0
votes
0answers
17 views

Cartier dual of an exact sequence

Suppose we have an exact sequence of affine finite flat commutative group schemes over an arbitrary ring $R$: \begin{equation} 0\rightarrow H\xrightarrow{i} G\xrightarrow{j} K\rightarrow 0 ...
4
votes
0answers
52 views

Defining the quantum group $U_q(\mathfrak{sl}_2)$

I've seen two defining relation for $U_q(\mathfrak{sl}_2)$ by the Serre relations $$[H,E]=E,\quad[H,F]=-F, \quad [E,F]=\frac{q^H-q^{-H}}{q-q^{-1}}, $$ or by taking $K=q^H$ $$KK^{-1}=K^{-1}K=1,\quad ...
2
votes
1answer
44 views

What does the notation $U(\frak{g})[[\hbar]]$ mean?

I'm reading the following motivation for studying quantum group but I'm unfamiliar with the double bracket notation in $$U(\frak{g})[[\hbar]].$$ Is this a special set of polynomials with coefficient ...
0
votes
0answers
40 views

Extension of homomorphism in multiplier algebra

I'm reading Timmermann's "An invitation to quantum groups and duality" and I have a problem in understanding following theorem: Every non-degenerate homomorphism $\phi : A \rightarrow M(B)$extends ...
1
vote
0answers
52 views

Example of Hopf categories and conatural transformation

Let $\mathcal{C}$ be a category. It is well known how to internalize the notion of category. Let $(C_0,C_1)$ be an internal category, with source $s$, target $t$, composition $c$ and unit $e$. One can ...
1
vote
0answers
42 views

Comultiplication in graded Hopf Algebras

Let $H$ be a graded Hopf algebra over some commutative ring $k$. I'm looking for a proof of the following result, which seems to be stated in various locations. For $x$ in $H$ of degree $n$ ...
0
votes
0answers
20 views

Reference request: product in $\mathbb{C}_q[X] \otimes \mathbb{C}_q[Y]$.

Let $a \otimes b, a'\otimes b' \in \mathbb{C}_q[X] \otimes \mathbb{C}_q[Y]$, where $X, Y$ are two algebraic varieties. Suppose that algebraic group $T$ acts on $X, Y$. Then there are coactions ...
1
vote
1answer
27 views

Is the coaction $\delta: A \to H \otimes A$ injective?

Let $A$ be an algebra and let $H$ be a bialgebra. Suppose that $A$ is an $H$-comodule. Then we have a coaction $\delta: A \to H \otimes A$. Is the coaction $\delta: A \to H \otimes A$ always ...
1
vote
0answers
58 views

Cocommutative k-Hopf algebra finite as k-vector space represents a constant group functor

I am working through some of my first exercises regarding Hopf algebras and I am kind of stuck with this one: Given an algebraically closed field $k$ and a cocommutative $k$-Hopf algebra $A$ finite ...
1
vote
1answer
36 views

Describe invariants using coaction.

Let $X$ be an algebraic variety and $G$ be an algebraic group which acts on $X$. We know that an invariant function $f$ in the coordinate ring $\mathbb{C}[X]$ is a function such that $g(f) = f$ for ...
1
vote
1answer
27 views

Coaction of a product.

Let $G$ be a group and let $X$, $Y$ be two algebraic varieties on which $G$ acts. Let $\delta_1: \mathbb{C}[X] \to \mathbb{C}[G] \otimes \mathbb{C}[X]$ be a coaction given by $\delta_1(f) = \sum ...
3
votes
0answers
35 views

Tensor product via the diagonal action of a Hopf algebra

Let $H$ be a Hopf algebra and $V$ and $W$ two left $H$-modules, then $V\otimes W$ is also a left $H$-module via the comultiplication of $H$. I now want to consider the functor $-\otimes_H (V\otimes ...
1
vote
0answers
24 views

Quotients and Hopf algebras

Let $A,B$ be $k$-Hopf algebras, for an algebraically closed field $k$. It follows that $h^A = $Hom$(A,-)$ and $h^B = $Hom$(B,-)$ are group functors. Now consider a natural transformation $\phi: h^A ...
0
votes
1answer
54 views

Frobenius map and Hopf algebras

I was wondering if I can get some help understanding a problem. Namely, consider the Frobenius map of $\mathbb{F}_p$-algebra $f:A \rightarrow A$ given by $a \mapsto a^p$. I showed that if $A$ is an ...
1
vote
1answer
60 views

Group objects in the category of rings

Are there group objects in: $\text{Ring}$ $\text{CRing}$ If so, why doesn't anyone talk about them? On the other hand, $$ \begin{align} cogroup \ objects \ in \ \text{CRing} &= co(group \ ...
6
votes
1answer
95 views

Is $\mathbb{H}P^\infty$ an H-space or not?

$\mathbb{R}P^\infty$ is H-space. $\mathbb{C}P^\infty$ is H-space. Is $\mathbb{H}P^\infty$ an H-space or not?
0
votes
1answer
74 views

Hopf bifurcation and limit cycles

$$dV/dt=10(V-\frac{V^3}{3}-R+I_{input})$$ $$dR/dt=0.8(-R+1.25V+1.5)$$ Use $I_{input}$ as the relative parameter to prove that there these equations undergo 2 hopf bifurcations and indicate whether ...
2
votes
2answers
53 views

Do coalgebras arise outside the study of bi/Hopf-algebras?

Hopefully the title is fairly self explanatory. I'm curious as to whether the coalgebra structure (that is, a vector space with a comultiplication and counit) comes up an any area of mathematics not ...
0
votes
0answers
16 views

How to show that action of an algebra $H$ on a vector space is the same as the coaction of $H^*$ on $V$?

Let $H$ be a Hopf algebra and $V$ a finite dimensional $H$-module. How to show that action of an algebra $H$ on a vector space is the same as the coaction of $H^*$ on $V$? Thank you very much.
2
votes
2answers
58 views

What is $\Delta(1)$ for $1$ in $U(\mathfrak{g})$?

Let $\mathfrak{g}$ be a semisimple Lie algebra and $U(\mathfrak{g})$ its universal enveloping algebra. Then $U(\mathfrak{g})$ is a hopf algebra. Is $\Delta(1) = 1 \otimes 1$ or $\Delta(1) = 1 \otimes ...
1
vote
0answers
17 views

Hopf Algebras Arising From Fourier Transforms?

At least for nice functions, a Fourier series is just a Laurent series on a circle (which means just substituting $z = re^{i\theta}$ into a Laurent series), so I can see Fourier analysis on Abelian ...
0
votes
1answer
43 views

The comultiplication on $\mathbb{C} Q $ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=Q$ the ...
2
votes
1answer
116 views

The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, ...
2
votes
1answer
57 views

Identities that connect antipode with multiplication and comultiplication

The group algebra $k(G)$ of any group $G$ satisfies as a Hopf algebra the following identities: $$ S\otimes S\circ \varDelta=\sigma\circ\Delta\circ S $$ $$ \nabla\circ S\otimes ...
1
vote
0answers
31 views

Isomorphism of affine group schemes of rank 2

I was reading Waterhouse's book "Introduction to Affine Group Schemes", when I found, in Chapter 2, an exercise about classification of Affine Group Schemes of Rank 2. I proved essentially that every ...
0
votes
1answer
33 views

Induced actions.

Let $U$ be the subgroup of $GL_n$ consisting of all unipotent upper triangular matrices. Suppose that there is an action of $U$ on a variety $X$. Let $\mathbb{C}[U]$ be the group algebra of $U$ and ...
1
vote
1answer
57 views

Is cocycle condition necesarry for coassociativity of coproduct and why is it called “cocycle condition”?

Let $\Delta_0$ and $\Delta$ be coproducts related by \begin{equation} \Delta h = \mathcal F \Delta_0 h \mathcal F^{-1} \end{equation} where $\mathcal F \in H\otimes H$ is Drinfeld twist and $h \in H$ ...
3
votes
0answers
57 views

Recovering a restricted Lie algebra from its restricted enveloping algebra

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $\mathfrak{g}$ be a restricted Lie algebra, with restricted enveloping algebra $u(\mathfrak{g})$. We can place a Hopf ...
3
votes
1answer
50 views

Are there always nontrivial primitive elements in a Hopf algebra?

Let $k$ be an algebraically closed field of arbitrary characteristic. Let $H$ be a Hopf algebra over $k$. We say $x\in H$ is a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$, where $\Delta$ ...
1
vote
1answer
53 views

Natural map of extension groups

Let $\Lambda$ be a cocommutative Hopf algebra over a commutative ring $R$. For two left $\Lambda$-modules $M$ and $N$, interpret $\mathrm{Ext}_{\Lambda}^n(M,N)$ as the set of equivalence classes of ...
1
vote
2answers
88 views

co-idempotents: algebraic dual of an idempotent element?

So many times you can write out the axioms for an algebraic structure (say an algebra over a ring) as commutative diagrams and then reverse all the arrows and get a new structure: say a coalgebra. ...
3
votes
0answers
96 views

Integral Homology of $BU$

We know that the integral cohomology of $BU$, $H^*(BU) = Z[c_1,c_2,...]$ where $|c_i| = 2i$ is the $i$th Chern class, with coproduct $\Delta c_i = \Sigma c_j\otimes c_{i-j}$. And at almost ...