The rational unipotent completion of a group 
is the universal pro-unipotent algebraic
group 
over 
equipped with a group homomorphism 
. Every group
homomorphism 
for a finite-dimensional unipotent algebraic
group 
over 
factors uniquely through a morphism of algebraic groups 
. Its canonical 
-step quotient is universal for targets of nilpotency class
at most 
.
Unipotent Completion
See also
Fundamental Group, Nilpotent Group, Quasi-Albanese Variety, Unipotent GroupExplore with Wolfram|Alpha
References
Hain, R. and Matsumoto, M. "Weighted Completion of Galois Groups and Galois Actions on the Fundamental Group of
." Compositio Math. 139,
119-167, 2003. https://doi.org/10.1023/B:COMP.0000005077.42732.93.
Cite this as:
Weisstein, Eric W. "Unipotent Completion." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UnipotentCompletion.html