Lattices in nilpotent Lie groups

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SUMMARY

This discussion focuses on the relationship between simply-connected nilpotent Lie groups and their lattices, specifically addressing the correspondence between Lie algebras and groups through exponential and logarithmic maps. It confirms that an ideal in the Lie algebra L corresponds to a normal subgroup in the Lie group G. Additionally, it explores the possibility of finding a normal subgroup in the lattice H that corresponds to a normal subgroup P in G. The construction of a Riemannian metric on the Lie group G related to its lattice H is also queried.

PREREQUISITES
  • Understanding of nilpotent Lie groups
  • Familiarity with Lie algebras and their properties
  • Knowledge of exponential and logarithmic maps in the context of Lie theory
  • Basics of Riemannian geometry
NEXT STEPS
  • Study the correspondence between ideals in Lie algebras and normal subgroups in Lie groups
  • Research the construction of Riemannian metrics on Lie groups
  • Explore examples of simply-connected nilpotent Lie groups and their lattices
  • Investigate the implications of normal subgroups in the context of group theory
USEFUL FOR

Mathematicians, particularly those specializing in algebraic topology, differential geometry, and representation theory, as well as graduate students studying Lie groups and algebras.

ibond
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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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1 a) is clearly true
 

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