# 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.

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### Find all invariant symmetric bilinear forms of $\mathfrak {sl}(2)$

Find all invariant symmetric bilinear forms of $\mathfrak {sl}(2)$ (as defined in the previous exercise; assume that the field $k$ is of characteristic zero.) Here is the exercise before it: Let $L$ ...
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### Determining the grouplike elements of a Hopf algebra

Here is the question I am trying to solve: For any Lie algebra $L$ determine the group of grouplike elements of the Hopf algebra $U(L).$ Some definitions: 1-To any Lie Algebra $L$ we assign an (...
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### Show that $HK^+$ is a coideal where $K$ is a subHopfalgebra of $H$, $K^+=\mathrm{ker}(\epsilon)\cap K$.

Show that $HK^+$ is a coideal where $K$ is a subHopfalgebra of $H$, $K^+=\mathrm{ker}(\epsilon)\cap K$. Because $\epsilon(ha)=\epsilon(h)\epsilon(a)=0$ for $a\in K^+$, $\forall h \in H$. What we need ...
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### Action on associated graded algebra inducing action on filtered algebra

Suppose $Q$ is a filtered algebra, with associated graded algebra $\text{gr}(Q)$. If we have an action of a ring $R$ on $\text{gr}(Q)$ (i.e. $\text{gr}(Q)$ is an $R$-module) then it seems clear that, ...
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### Why can't you consider coideal generated by sets.

The therm "coideal generated by a set" don't exist in literature but didn't found anything explaining why, so i formulated an example of a 6-dimensional coalgebra in wich there's a 1-...
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### Why we need an orthonormal basis?

Here is the question I am trying to solve: Prove that $\lambda$ is injective. Here is the definition of the linear map $\lambda$: Let $f: U \to U'$ and $g: V \to V'$ be linear maps. We define their ...
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