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1¡¡introduction
¡¡1.1 outline of the book
¡¡1.2 suggestions for further reading
¡¡1.3 notations, background, conventions
2 bilinear forms, quadratic forms and their isometry groups.
¡¡2.1 standard results on quadratic forms and reflections
¡¡¡¡2.1.1 principal ideal domains (pids)
¡¡2.2 linear algebra
¡¡¡¡2.2.1 interpretation of nonsingularity
¡¡¡¡2.2.2 extension of scalars
¡¡¡¡2.2.3 cyclicity of the values of a rational bilinea.r form
¡¡¡¡2.2.4 gram matrix
¡¡2.3 discriminant group
¡¡2.4 relations between a lattice and sublattices
¡¡2.5 involutions on quadratic spaces
¡¡2.6 standard results on quadratic forms and reflections, ii
¡¡¡¡2.6.1 involutions on lattices
¡¡2.7 scaled isometries: norm doublers and triplers
3 general results on finite groups and invariant lattices
¡¡3.1 discreteness of rational lattices
¡¡3.2 finiteness of the isometry group
¡¡3.3 construction of a g-invariant bilinear form
¡¡3.4 semidirect products and wreath products
¡¡3.5 orthogonal decomposition of lattices
4 root lattices of types a, d, e
¡¡4.1 background from lie theory
¡¡4.2 root lattices, their duals and their isometry groups
¡¡¡¡4.2.1 definition of the an lattices
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