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Questions tagged [hopf-algebras]

For questions about Hopf algebras and related concepts, such as quantum groups. The study of Hopf algebras spans many fields in mathematics including topology, algebraic geometry, algebraic number theory, Galois module theory, cohomology of groups, and formal groups and has wide-ranging connections to fields from theoretical physics to computer science.

Hopf algebras, named after Heinz Hopf, was first introduced in the theory of algebraic topology, while studying cohomology of Lie groups, but in recent years has been developed by many mathematicians and applied to other areas of mathematics such as algebraic groups, combinatorics, mathematical physics and Galois theory.

It is a structure that is simultaneously a (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism, called antipode, satisfying a certain property.

The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.

There is a wide variety of variations of the notion of Hopf algebra, relaxing properties or adding structure. Examples are weak Hopf algebras, quasi-Hopf algebras, (quasi-)triangular Hopf algebras, quantum groups, hopfish algebras etc.

For more details you may find the following references:
$1.~~$ "Introduction to Hopf algebras and representations" by Kalle Kytola
$2.~~$ "Hopf Algebras in Combinatories" by Darij Grinberg & Victor Reiner
$3.~~$ "Hopf Algebra" from Wikipedia
$4.~~$ "A Very Basic Introduction to Hopf Algebras" by J.M. Selig

522 questions
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Do we have $\epsilon(h)=\epsilon(h_{(1)})\epsilon(h_{(2)})$?

Let $H$ be a Hopf algebra and $\epsilon: H \to \mathbb{C}$ the counit. Do we have $\epsilon(h)=\epsilon(h_{(1)})\epsilon(h_{(2)})$? I think that $h = \epsilon(h_{(1)})h_{(2)}$. Therefore $\epsilon(h) = \epsilon(\epsilon(h_{(1)})h_{(2)}) =…
LJR
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What does it mean for $\operatorname{Ad}_g(-)$ to be trivial on commutative Hopf algebra $H$?

Just curious, what does it mean for the adjoint action to be trivial on a commutative Hopf algebra $H$? Does it mean $\operatorname{Ad}_g(f)=\epsilon(g)f$, where $\epsilon$ is the counit, or $\operatorname{Ad}_g(f)=f$? If $H$ is commutative as an…
krolhauser
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Are antipodes of free, finite rank Hopf algebras over general rings invertible?

It is a well-known result by Larson and Sweedler that, for finite-dimensional Hopf algebras over a field, the antipode is always a linear isomorphism. My question is whether this property still holds for free, finite rank Hopf algebras over…
Minkowski
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How to prove the following results for $H$-bimodule $V$?

Suppose $(H, \cdot, \eta, \Delta,\varepsilon,S)$ be a Hopf algebra, and $V$ be a (finite-dimensional) $H$-bimodule. Then, how can we prove $\alpha\dot{}h-h.\alpha = 0$ for all $h \in H$ if we have $\alpha \in V$ which satisfy…
ChoMedit
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Let B be a bialgebra. Show that the following are equivalent

Let B be a bialgebra. Show that the following are equivalent: B is a Hopf algebra. The maps $T_1$, $T_2$: $B \otimes B \to B \otimes B$ determined by $T_1(a \otimes b) = \sum a_{(1)} \otimes a_{(2)} b$ and $T_2(a \otimes b) = \sum a b_{(1)} \otimes…
fosterc4
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How Do I Prove that $\gamma(f\star g)=\gamma(f)\star\gamma(g)$ for the Following Proposition?

Let $C$ and $C'$ be two $\mathbf{K}$-algebras, and let $A$ and $A'$ be two $\mathbf{K}$-algebras. Let $\gamma\colon C\to C'$ be a $\mathbf{K}$-algebra morphism. Let $\alpha\colon A\to A'$ be a $\mathbf{K}$-algebra morphism. The…
webby
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In a Hopf algebra, is $u\epsilon (h) \otimes h = h$?

Let $(H, m, u, \Delta, \epsilon, S)$ be a Hopf algebra and $h\in H$. Do we have the equality $$u\epsilon(h)\otimes h = h$$ Or maybe put $1\otimes h$ on the RHS or something else "isomorphic"? EDIT: To give the context: this is what the exercise asks…
ploosu2
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Understanding homomorphism from coalgebra to algebra

Could someone please explain what exactly a homomorphism from coalgebra to algebra (from this paper: 1, page 10, definition 5.1). I understand a homomorphism as a map between two structures which preserves operations and their neutral elements, but…
Ordev Agens
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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.
LJR
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