---
date: 2021-10-21 18:42
modification date: Saturday 23rd October 2021 21:58:45
title: algebraic group
aliases: [algebraic groups, isogeny]

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Last modified: `=this.file.mday`

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- Tags: 
	- #todo/untagged
- Refs:
	- [attachments/alggroups.pdf](attachments/alggroups.pdf)
- Links:
	- [reductive](reductive)
	- [adele](Unsorted/adelic%20group.md)

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# algebraic group

- **Algebraic group**: usually means an affine algebraic group scheme.
- **Isogeny**: surjective morphism of algebraic groups with finite kernel.
- **Simply connected** iff $G$ is semisimple and every isogeny $H\to G$ with $H$ connected is an isomorphism.
- **Radical**: largest normal solvable subgroup, $R_G$.
- **Unipotent radical**: $R_u G = R_G \intersect U(G)$ where $U(G)$ are the unipotent elements.
- **Reductive**: 
	- The unipotent radical vanishes.
	- Implies every rep decomposes into irreducibles.
- **Levi**: any $H$ such that $R_u G \semidirect H = G$.
- **Borel**: maximal connected solvable subgroup.
- **Parabolic**: $P\leq G$ with $G/P$ projective.
	- Stablizer of a flag, not necessarily complete.
	- Equivalently $P\contains B$ a Borel


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