All Questions
Tagged with hopf-algebras algebraic-groups
25
questions
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43
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Affine group schemes
I am trying to clarify the definition of affine group schemes. It can be seen as a representable functor from the category of algebras to the category of groups.
My question
Let $k$ be an ...
- 11
1
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0
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28
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Group isomorphism $\operatorname{Tgt}_e(G) \cong \operatorname{Hom}_{k-\text{linear}}(\ker(\epsilon)/\ker(\epsilon)^2, k)$ for algebraic groups.
I'm reading from Milne's text https://www.jmilne.org/math/CourseNotes/RG.pdf, in chapter 8 on Lie algebras of (affine) algebraic groups $G$ over $k$. In it, he claims in 8.6 that there is an ...
1
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1
answer
74
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Which algebraic subvarieties of a group variety have a group structure?
Let $G$ be an algebraic group. Given an algebraic subvariety $X\subseteq G$, is there any way simple criterion to determine whether $X$ has a group structure or not?
For example, when $X$ and $G$ are ...
- 1,260
2
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1
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146
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Are there exotic group laws on affine space in characteristic zero?
Let $k$ be a field of characteristic zero, and let $X = \mathbb{A}^n_k$ be affine $n$-space. Up to $k$-rational translation, we may assume that a map $e : \operatorname{Spec} k \to X$ is the inclusion ...
- 1,601
3
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0
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66
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Regular functions on torsors
Let $X \rightarrow Y$ be a torsor for a linear algebraic group $G$ (i.e. $X$ is a principal $G$-bundle over $Y$). Assume also that both $X$, and $Y$ are affine. What can be said about $\mathbb{C}[X]$ ...
- 135
3
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69
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When is an image of a Hopf algebra a Hopf algebra?
Suppose $R\subseteq S$ is a flat extension of rings and $A/R$, $B/S$ are flat Hopf algebras. Let $\varphi:A\otimes_R S\to B$ be a surjective $S$-Hopf algebra homomorphism. When is it the case that $\...
- 593
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Coordinate algebra of image of homomorphism between linear algebraic groups
Let $f: G\to H$ be a homomorphism of linear algebraic groups. Let $f^*: k[H]\to k[G]$ be the corresponding Hopf algebra morphism. Then $f^*$ factors as $k[H] \twoheadrightarrow k[H]/I\hookrightarrow ...
- 2,438
2
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0
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55
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For an algebraic group $G$, a $G$-module induces a $Dist(G)$-module
I'm reading Representation of Algebraic Groups of Jantzen about Distribution of Algebra. In chapter 7, page 103, 7.11, the author is stating that if $G$ is a group scheme over $k$, then any $G$-module ...
- 4,845
1
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1
answer
173
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Character group of diagonalizable group scheme
Suppose $k$ is an integral domain, $\Lambda$ is a commutative group, $k[\Lambda]$ is the corresponding group ring and $\text{Diag}(\Lambda) = \text{Spec}(k[\Lambda])$ is the diagonalizable group ...
- 4,845
2
votes
1
answer
157
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Lie operator is left exact
In page 190, part (c) of the book "Algebraic groups, the theory of group schemes of finite type over a field" of Milne, there's a part stating that:
An exact sequence of algebraic groups $e ...
- 4,845
6
votes
1
answer
431
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Counterexample to "kernel determines image"
Working over a base field, there is a typical homomorphism theorem for affine algebraic groups ensuring that any two homomporphisms $G \to H_1$, $G \to H_2$ with the same kernel in $G$ have isomorphic ...
- 7,437
5
votes
2
answers
396
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Distributions of a group scheme as differential operators.
I'm trying to understand the distributions on an affine group scheme act as differential operators. Let $X$ be a scheme and $G$ a group scheme both affine over some commutative ring $k$. Suppose $\...
2
votes
1
answer
130
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Prove that $\Delta(x) = x \otimes1 + 1 \otimes x$ mod $I \otimes I $ for all $x \in I$.
I got stuck on this problem, so if anyone can give me a hint on this, I really appreciate.
Let $I$ be the augmentation ideal in Hopf algebra $A$. Prove that $\Delta(x) = x \otimes1 + 1 \otimes x$
mod $...
- 4,845
0
votes
0
answers
45
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Question about the equivalence between the categories of affine group schemes and Hopf algebras over $k$.
There is a proof given here (Theorem 1.3.4) of the equivalence between the categories $\mathbf{AffGrpSch}/k$ and $\mathbf{HopfAlg}^{opp}_k$.
The author writes in the line before the theorem that ...
- 2,072
2
votes
1
answer
438
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Book recommendation for Hopf algebras [duplicate]
My motivation for studying Hopf algebras is twofold:
I'm interested in representation theory and the algebraic structure of group rings.
I will also take a course on algebraic groups soon and I ...
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