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De Medts T. Linear algebraic groups

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De Medts T. Linear algebraic groups
Ghent: University of Ghent, 2019. — 145 p.
Contents :
Preface
Introduction
First examples
The building bricks
Finite algebraic groups
Abelian varieties
Semisimple linear algebraic groups
Groups of multiplicative type and tori
Unipotent groups
Extensions
Solvable groups
Reductive groups
Disconnected groups
Overview
Algebras
Definitions and examples
Tensor products
Tensor products of K-modules
Tensor products of K-algebras
Categories
Definition and examples
Functors and natural transformations
The Yoneda Lemma
Algebraic geometry
Affine varieties
The coordinate ring of an affine variety
Affine varieties as functors
Linear algebraic groups
Affine algebraic groups
Closed subgroups
Homomorphisms and quotients
Affine algebraic groups are linear
Jordan decomposition
Jordan decomposition in GL(V )
Jordan decomposition in linear algebraic groups
Lie algebras and linear algebraic groups
Lie algebras
The Lie algebra of a linear algebraic group
Topological aspects
Connected components of matrix groups
The spectrum of a ring
Separable algebras
Connected components of linear algebraic groups
Dimension and smoothness
Tori and characters
Characters
Diagonalizable groups
Tori
Solvable linear algebraic groups
The derived subgroup of a linear algebraic group
The structure of solvable linear algebraic groups
Borel subgroups
Semisimple and reductive groups
Semisimple and reductive linear algebraic groups
The root datum of a reductive group
Classification of the root data
References
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