Outer automorphisms of classical algebraic groups
| Autor: | Anne Quéguiner-Mathieu, Jean-Pierre Tignol |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Classical group
Pure mathematics Degree (graph theory) Group (mathematics) General Mathematics 010102 general mathematics Outer automorphism group Group Theory (math.GR) Type (model theory) Automorphism 01 natural sciences Simple (abstract algebra) Algebraic group 0103 physical sciences FOS: Mathematics 010307 mathematical physics 0101 mathematics 20G15 11E57 Mathematics - Group Theory Mathematics |
| Popis: | The so-called Tits class, associated to an adjoint absolutely almost simple algebraic group, provides a cohomological obstruction for this group to admit an outer automorphism. If the group has inner type, this obstruction is the only one. In this paper, we prove this is not the case for classical groups of outer type, except for groups of type $^2\mathsf{A}_n$ with $n$ even, or $n=5$. More precisely, we prove a descent theorem for exponent $2$ and degree $6$ algebras with unitary involution, which shows that their automorphism groups have outer automorphisms. In all other relevant classical types, namely $^2\mathsf{A}_n$ with $n$ odd, $n\geq3$ and $^2\mathsf{D}_n$, we provide explicit examples where the Tits class obstruction vanishes, and yet the group does not have outer automorphism. As a crucial tool, we use "generic" sums of algebras with involution. |
| Databáze: | OpenAIRE |
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