This entry is about the notion of lattice in group theory/quadratic form-theory. For other notions see at lattice (disambiguation).
Classically, a lattice in the Cartesian space is a discrete subgroup (of the underlying topological abelian group) that spans as a vector space over . This may be generalized, from to a general locally compact abelian group.
A lattice in a locally compact Hausdorff abelian group is a subgroup that is discrete and cocompact, meaning that the quotient group with the quotient topology is compact.
Applying Pontryagin duality, the dual of the quotient map is in that case a discrete subgroup that is also cocompact (its cokernel being the compact group ). This is called the dual lattice of .
Notable examples of classical lattices (in ) include
The standard diagonal inclusion of a global field (such as a number field) into its ring of adeles is a lattice in the more general sense. Recalling that is Pontryagin dual to itself, the lattice is identified with its dual lattice.
Peter Engel, Louis Michel, Marjorie Senechal, Lattice Geometry, IHES/P/04/45 (2004) pdf, cds:859509
Andries E. Brouwer, Lattices, Course notes (2002) [pdf, pdf]
John Conway, N. Sloane, Low dimensional lattices I: Quadratic forms of small determinant
Wikipedia, Lattice (group)
Wikipedia, Lattice (module)
Applications in heterotic string theory and F-theory:
Last revised on June 11, 2024 at 14:34:47. See the history of this page for a list of all contributions to it.