free Lie algebra
Fix a set and a commuative unital ring . A free -Lie algebra on is any Lie algebra together with an injection such that for any -Lie algebra and function implies the existance of a unique Lie algebra homomorphism where . This universal mapping property is commonly expressed as a commutative diagram:
To construct a free Lie algebra is generally and indirect process. We begin with any free associative algebra on , which can be constructed as the tensor algebra over a free -module with basis . Then is a -Lie algebra with the standard commutator bracket for .
Now define as the Lie subalgebra of generated by .
It is generally not true that . For example, if then but is not in .
- 1 Nathan Jacobson Lie Algebras, Interscience Publishers, New York, 1962.
|Title||free Lie algebra|
|Date of creation||2013-03-22 16:51:11|
|Last modified on||2013-03-22 16:51:11|
|Last modified by||Algeboy (12884)|
|Defines||free Lie algebra|