A lattice in a Lie group is a discrete subgroup such that the quotient of the Lie group by the lattice has finite volume with respect to the Haar measure. Lattices are crucial for understanding the structure and representation of Lie groups, as well as for applications in geometry and number theory.

Lattice In Lie Groups

lattice in a Lie group

A discrete subgroup is a subset of a topological group that is itself a group and is endowed with the discrete topology, meaning its elements are isolated points. This concept is crucial in the study of group actions on manifolds and has significant implications in areas such as geometry, number theory, and the theory of Lie groups.

discrete subgroup

quotient of the Lie group

The finite volume method is a numerical technique used for solving partial differential equations by dividing the domain into discrete volumes and applying conservation laws. It is particularly effective for fluid dynamics problems because it ensures the conservation of fluxes across the boundaries of each control volume.

finite volume

Haar measure is a way to define a translation-invariant measure on locally compact topological groups, allowing integration over the group in a manner that respects its group structure. It is fundamental in harmonic analysis, representation theory, and the study of symmetries in mathematical physics, providing a rigorous foundation for integrating functions over groups like the circle, real numbers, or more complex structures such as Lie groups.

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