Lattices in nilpotent Lie groups

L and G. b) However, the converse is not always true, as there may not necessarily be a one-to-one correspondence between normal subgroups in G and H.In summary, there is a one-to-one correspondence between the Lie algebra L of a simply-connected nilpotent Lie group G and G itself through the exp and log maps. This means that to an ideal in L, there corresponds a normal subgroup in G. However, the converse is not always true, as there may not necessarily be a one-to-one correspondence between normal subgroups in G and H. Additionally, the construction of Riemann metric on G corresponding to its lattice H is not always possible due
  • #1
ibond
1
0
Please, help me with the following questions or recommend some good books.

1) We have a simply-connected nilpotent Lie group G and a lattice H in G. Let L be a Lie algebra of G. There is a one to one correspondence between L and G via exp and log maps.
a) Is it true, that to an ideal in L corresponds a normal subgroup in G?
b) If we have a normal subgroup P in G, can we find a normal subgroup in H with Lie group P?

2) What is the construction of Riemann metric on the Lie group G corresponding to its lattice H?
 
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  • #2
1 a) is clearly true
 

What is a lattice in a nilpotent Lie group?

A lattice in a nilpotent Lie group is a discrete subgroup that is isomorphic to the group itself. It is a discrete and cocompact subgroup, meaning it is both discrete (has no accumulation points) and covers the entire group.

What is the significance of lattices in nilpotent Lie groups?

Lattices in nilpotent Lie groups play an important role in mathematics and physics, particularly in number theory and geometry. They have applications in coding theory, cryptography, and the study of manifolds and geometric structures.

How are lattices in nilpotent Lie groups related to crystallography?

In crystallography, lattices in nilpotent Lie groups correspond to the symmetry groups of crystal structures. The structure of crystals can be described using the concept of a lattice, and lattices in nilpotent Lie groups provide a mathematical framework for studying these structures.

What are some examples of nilpotent Lie groups with lattices?

Some examples of nilpotent Lie groups with lattices include the Heisenberg group, the 3-dimensional unimodular group, and the 7-dimensional group with Lie algebra spanned by 3-step nilpotent matrices.

What are some open questions and conjectures related to lattices in nilpotent Lie groups?

There are still many open questions and conjectures related to lattices in nilpotent Lie groups, such as the existence of lattices in certain classes of nilpotent Lie groups, the classification of lattices in certain dimensions, and the relationship between lattices and other geometric structures on nilpotent Lie groups.

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