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Questions tagged [hopf-algebras]

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 structure exists on the group algebra $k G$ of a finite group $G$.

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Explicit examples of finite dimensional, involutive Hopf algebras with traceless antipode?

$\require{AMScd}$ In the paper [1], it is shown that there exist finite dimensional, semisimple Hopf algebras $H$ where the antipode $S:H \to H$ is traceless. Unfortunately, the method of proof in [...
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Functoriality of the Hopf dual

Given Hopf $\mathbb{C}$-algebra $H$, it's Hopf dual $H^o$ is the largest Hopf algebra contained in $H^*$, the $\mathbb{C}$-linear dual of $H$. (This is well known to be well-defined, see for example ...
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It there a nice way to describe the structure of Malcev-complete groups?

Let $\mathbb k$ be a field of characteristic zero. The grouplike functor $\mathbb G$ from complete Hopf algebras to groups is a faithful functor. Its image is the category of Malcev-complete groups ...
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Is there another quantum deformation of sl(2)?

By looking at defining relations of standard deformation of $\mathfrak{sl}_2$, which are: $$ [E,F] = \frac{q^{H}-q^{-H}}{q-q^{-1}}, \quad [H,E] = 2E, \quad \text{ and } \quad [H,F] = -2F, $$ some ...
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Modules over Hopf Algebras and $E_2$-algebras

Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E_2$-algebras: https://arxiv.org/pdf/1309.2820.pdf I was wondering if anybody knows of a nice relationship between ...
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How are the unit/counit of a Hopf algebra and of an categorical adjunction related?

For categories $\mathcal{C}$ and $\mathcal{D}$, a pair of functors $\mathcal{C} \xrightarrow{\;L\;} \mathcal{D}$ and $\mathcal{C} \xleftarrow{\;R\;} \mathcal{D}\,$ are an adjoint pair if we have ...
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Independence of characters with respect to polynomials

I came across the following property : Let $\mathfrak{g}$ be a Lie algebra over a ring $k$ without zero divisors, $\mathcal{U}=\mathcal{U}(\mathfrak{g})$ be its enveloping algebra. As such, $\...
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Lie algebra of a compact Lie group

Let $\mathcal{G}$ be a compact (real) Lie group. We know that the Lie algebra $\mathfrak{g}$ of $\mathcal{G}$ is, by definition, the space of all left-invariant (smooth) vector fields over $\mathcal{G}...
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Tannaka-Krein reconstruction and rigidity

Let $\mathcal{C}$ be a rigid monoidal category together with a quasi-monoidal functor $\omega:\mathcal{C}\to\mathsf{vec}_{\Bbbk}$ to finite-dimensional vector spaces over a field $\Bbbk$, i.e. we have ...
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Hopf algebra interpretation of hypergraph duality?

The work of Aguiar and Ardila (https://arxiv.org/abs/1709.07504) on Hopf monoids for generalized permutohedra gives a Hopf monoid structure on the collection of hypergraphs; see sections 19 and 20 of ...
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Lang's Jacobian identity: slicker, elementary proof?

In Jeffrey Lang, A Jacobian identity in positive characteristic, J. Commut. Algebra, Volume 7, Number 3 (2015), pp. 393--409, the following result is proven: Theorem 1. Let $p$ be a prime. Let $\...
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Hopf dual of the Hopf dual

Given any Hopf algebra $A$ over a field $k$, one can also define the Hopf dual $A^*$ of as follows: Let $A^∗$ be the subspace of the full linear dual of $A$ consisting of elements that vanish on some ...
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The Ungraded Milnor-Moore Theorem

Let $k$ be a field of characteristic $0$. There is a functor $U$ from Lie-algebras over $k$ to Hopf algebras over $k$ sending a $k$-Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\...
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When is this map of Hopf algebras Surjective?

I'm reading Akhil Mathew's blog post on Formal Lie Theory in Characteristic Zero. Let $H$ be cocommutative Hopf algebra over a field $k$. We can form $\mathfrak{g}$, the Lie algebra over $k$ ...
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Unbiased Hopf algebras

In category theory, a notion of monoidal category in which every sequence $X_1, \ldots , X_n$ ($n\ge 0$) of objects has a specified product is called an ``unbiased monoidal category'' (see Section 3.1 ...
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The dual of a dual in a rigid tensor category

For a rigid tensor category $\cal{C}$, can it happen that, for some $X \in {\cal C}$, we have that $X$ is not isomorphic to $(X^{*})^*$, for $*$ denoting dual? If so, what is a good example.
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Free module over $H$-module algebra

Let $H$ be a finite dimensional Hopf algebra, $R$ be a $H$-module algebra and $V$ be a finite dimensional $H$-module such that $R\otimes_{k} V$ is a finitely generated $R$ module under the action: $r.(...
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Quasi-symmetric generalizations of classical symmetric functions

I am looking for quasi-symmetric versions of the classical $e_\lambda$ and $h_\lambda$ (the elementary and complete homogeneous symmetric functions). Is there some reference for this? I am aware of ...
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Is there a ''simple'' formula for the inverse of the Drinfeld associator?

The Drinfeld associator $\Phi(x_0, x_1)$ encodes the parallel transport of the Knizhnik-Zamolodchikov (KZ) connection $\nabla$ on the bundle $\mathbb{C}\langle\langle x_0, x_1\rangle\rangle$ of formal ...
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Hopf Subalgebras of Quantized Algebras

As is well known, quantized enveloping algebras $U_q(\frak{g})$ admit far fewer sub-Hopf algebras than classical enveloping algebras $U(\frak{g})$. As one can check directly, for appropriate subsets ...
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What additional property does the antipode give on the category of all modules over an Hopf algebra?

It is well known that many constructions involving bialgebras extends to monoidal categories, and often becomes more natural in that framework. If one cares about the category of finite dimensional ...
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Module algebras and comodule algebras

Let $H$ be a Hopf algebra and $A$ an algebra. Let $H^*$ be the dual Hopf algebra of $H$. Then by Proposition 1.6.11 in the book Foundations of Quantum Group Theory by Shahn Majid, $A$ is a right $H$-...
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Exceptional Quantum Groups as FRT-Algebras

Let $\frak{g}$ be a simple Lie algebra of A,B,C,or D series type. Moreover, let $U_q(\frak{g})$ be its Drinfeld-Jimbo quantized enveloping algebra, and $G_q$ the quantized enveloping algebra. As is ...
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For any finite-dimensional Hopf C*-algebra, can one make the multiplication and co-multiplication cyclically symmetric simultanously?

For any finite-dimensional *-algebra, one can choose a basis such that the coefficients tensor of the anti-linear map $(a,b)\rightarrow (ab)^*$ becomes cyclically symmetric. (Any *-algebra is ...
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When is a Frobenius Algebra a Quasi-Frobenius Ring?

Let $F$ be Frobenius algebra in the monoidal category $\mathcal{C}$ of bimodules over a not-necessarily commutative algebra $A$. When is it true that $F$ is a quasi-Frobenius ring. For example, this ...
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On the isomorphism problem of enveloping algebras

Let $\mathfrak{g}$ and $\mathfrak{g}'$ be Lie algebras. It is known that if $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as associative algebras, then it is not necessarily true that $\mathfrak{g}\cong \...
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$Q(f+g)_*=Q(f_*+g_*)$ (The maps induced by the sum is the sum of induced maps modulo decomposables [Reference request]

Let $X, Y$, let's say, homotopy commutative $H$-spaces, $f,g$ maps from $X$ to $Y$. (Actually we only need $Y$ to be homotopy commutative $H$_space, but the statement is easier if we also suppose $X$ ...
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A diagram for understanding action/coaction compatibility in a Yetter-Drinfeld module

For a Hopf algebra $H$ with antipode $S$, let $M$ be a left $H$-module with the action $h \otimes m \mapsto \rho(h,m)$, and also a left $H$-comodule with coaction $\delta \colon m \mapsto m^{(-1)} \...
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2answers
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Are there examples of finite-dimensional weak Hopf C*-algebras with non-involutive antipode?

For finite-dimensional (non-weak) Hopf C*-algebras it is known that the antipode is always involutive, as claimed e.g. in https://arxiv.org/pdf/1007.5283.pdf. I couldn't find the same statement for ...
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Hopf-Galois extensions where the “extension” is a module?

For $H$ a Hopf-algebra, an $H$-Hopf-Galois extension is a map of rings $\phi\colon\thinspace A\to B$ such that $H$ coacts on $B$ over $A$, $B\otimes_AB\cong B\otimes H$, and the cofixed points, or the ...
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Group schemes over ring of Witt vectors and their representing algebras

Let $G$ be an affine groups scheme over $\mathbb Z$. As such it has an associated Hopf algebra, $A=\mathbb Z[G]$ such that $G(R)$ is naturally identified with the set $\hom_{Rng}(A,R)$ of ring ...
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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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The difference between $q$-deformations and $h$-deformations

What is the difference between $q$-deformations and $h$-deformations of universal enveloping algebras? In chapter XVI of Quantum groups by Kassel, a very precise definition of a quantum enveloping ...
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Quantum invariant: The canonical $2$-tensor

In Chapter XVI Kassel introduces a proper definition of a quantum universal enveloping algebra of a Lie algebra $\mathfrak{g}$. (See definition XVI.5.1). Notice that a quantum enveloping algebra has a ...
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Detecting skew-primitives in representation categories

Suppose $H$ and $H'$ are two (possibly infinity dimensional) Hopf algebras which are not isomorphic as Hopf algebras, but are isomorphic as algebras. More specifically they are not isomorphic as Hopf ...
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Relative version of Hopf cyclic cohomology

In this paper Connes and Moscovici introduced the Hopf algebra of transverse differential operators in order to compute the index formula for the diffeomorphism invariant geometry. They developed the ...
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Algebras for (Koszul) Hopf operads

If necessary, we can restrict the following to the case where we consider only Hopf (co)operads in the category of chain complexes over fields of characteristic zero. In case of ordinary operads, ...
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Inner automorphisms of Hopf algebras

Is there a reasonable notion of an inner automorphism of a Hopf algebra $H$ which in the case of a group ring $H=\mathbb KG$ for a group $G$ reduces to a conjugation by $g\in G$?
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What are the primitive elements in a polynomial Hopf algebra with primitive indeterminates?

Asked on math.stackexchange https://math.stackexchange.com/questions/2510606/what-are-the-primitive-elements-in-a-polynomial-hopf-algebra-with-primitive-inde but didn't get response (in fact got a ...
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Category of (co)commutative Hopf monoids in an exact category

I'm transferring this question over from SE, since it didn't get much attention over there. Let $(C, \otimes)$ be an exact monoidal category, and let $H(C)$ be the category of cocommutative and ...
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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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Looking for an automorphism constructed from Hopf algebraic data

I am in an automorphism quest. Let $H$ be a quasitrianglar Hopf Algebra with R-matrix $\mathcal{R} \in H \otimes H$. I know that $\mathcal{R}_{21}^{-1 }$ is a solution of the Yang Baxter equation and ...
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Stable homotopy type theory?

This is a combined question, strictly speaking I am asking three questions concerning, respectively, homotopy type theory, stable homotopy theory and Yetter-Drinfeld modules. But I believe in the ...
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1answer
171 views

Number of Isomorphism Classes of Corepresentations of A Compact Quantum Group

Given a compact quantum group $(G,\Delta)$, with dense Hopf algebra $H$, is it always true that, up to isomorphism, $H$ will have a countable number of irreducible comodules?
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Examples of representations of quantum groups

I am trying to learn some basic theory of quantum groups $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a simple Lie algebra, say $sl_n(\mathbb{C})$. As far as I heard the finite dimensional ...
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group action on Tor groups of modules and smash product

I am trying to understand theorem 3.4.2 from the paper "Bernstein-Gelfand-Gelfand complexes and cohomology of nilpotent groups over $\mathbf Z_{(p)}$ for representations with $p$-small weights" by ...
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Analog of Haar element in an algebra

In a Hopf algebra $H $ (over some field $ k $), there is the notion of a Haar element $ h \in H$. This is an element of the algebra which has the property that if $ V $ is a representation of $ H $, ...
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The order of the antipode in a Hopf algebra

As a result of Radford, any finite-dimensional Hopf algebra an antipode of finite order. My question: How can we classify all finite-dimensional Hopf algebras whose antipode is identity? Here are ...
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Categorical interpretation of quantum double $D(A,B,\eta)$

It is known that the Drinfel'd double $D(A)$ of a Hopf algebra $A$ is characterized by the following two properties: The category of left $D(A)$-modules $_{D(A)}\mathcal{M}$ is equivalent to the ...
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Is Sweedler's Hopf algebra factorizable?

Sweedler's 4-dimensional Hopf algebra admits a one-parameter family of triangular structures given by \begin{equation} R_{\lambda}:=1\otimes1+1\otimes g+g\otimes 1-g\otimes g-\frac{\lambda}{2}(x\...