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Geometric actions of classical algebraic groups

Geometric actions of classical algebraic groups
Geometric actions of classical algebraic groups
Let k be an algebraically closed field of arbitrary characteristic p. An affine algebraic group G is an affine algebraic variety over k with a group structure such that multiplication and inversion maps are morphisms of varieties. A special class of affine algebraic groups are the so called classical groups Cl(V), groups of isometries of a finite dimensional k-vector space V with respect to a certain form on V {e.g. a zero form, a symplectic form or a non-degenerate quadratic form. These groups are: GL(V) the general linear group, Sp(V) the symplectic group and O(V) the orthogonal group. Let G = Cl(V). Various (closed) subgroups H of G can be defined naturally in terms of the geometry of V {H may be the stabiliser of a subspace of V, or a direct sumdecomposition of V, or a non-degenerate form on V, for example. Let H be such a subgroup and let = G=H be the corresponding coset space. Then is a variety with a natural algebraic action of G. We define geometric subgroups of G to be the closed subgroups arising in this manner. Consequently, for H a geometric subgroup, we say that the natural action of G on = G=H is a geometric action. We define C (x) to be the set of points in fixed by x. Then C (x) is a subvariety, and we can show that dimC (x) = dim.
Rainone, Raffaele
bec8cb7d-ec53-4d3f-932b-c6205ce21c1b
Rainone, Raffaele
bec8cb7d-ec53-4d3f-932b-c6205ce21c1b
Kropholler, Peter
0a2b4a66-9f0d-4c52-8541-3e4b2214b9f4

Rainone, Raffaele (2014) Geometric actions of classical algebraic groups. University of Southampton, Mathematics, Doctoral Thesis, 357pp.

Record type: Thesis (Doctoral)

Abstract

Let k be an algebraically closed field of arbitrary characteristic p. An affine algebraic group G is an affine algebraic variety over k with a group structure such that multiplication and inversion maps are morphisms of varieties. A special class of affine algebraic groups are the so called classical groups Cl(V), groups of isometries of a finite dimensional k-vector space V with respect to a certain form on V {e.g. a zero form, a symplectic form or a non-degenerate quadratic form. These groups are: GL(V) the general linear group, Sp(V) the symplectic group and O(V) the orthogonal group. Let G = Cl(V). Various (closed) subgroups H of G can be defined naturally in terms of the geometry of V {H may be the stabiliser of a subspace of V, or a direct sumdecomposition of V, or a non-degenerate form on V, for example. Let H be such a subgroup and let = G=H be the corresponding coset space. Then is a variety with a natural algebraic action of G. We define geometric subgroups of G to be the closed subgroups arising in this manner. Consequently, for H a geometric subgroup, we say that the natural action of G on = G=H is a geometric action. We define C (x) to be the set of points in fixed by x. Then C (x) is a subvariety, and we can show that dimC (x) = dim.

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Published date: February 2014
Organisations: University of Southampton, Pure Mathematics

Identifiers

Local EPrints ID: 366485
URI: http://eprints.soton.ac.uk/id/eprint/366485
PURE UUID: 95cae9ca-9d91-4261-9941-55bc5e1caf73
ORCID for Peter Kropholler: ORCID iD orcid.org/0000-0001-5460-1512

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Date deposited: 15 Oct 2014 11:52
Last modified: 26 Jul 2019 00:31

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