Universal Property
Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → UL, (notation as above) we say that U is the universal enveloping algebra of X if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: X → AL there exists a unique unital algebra homomorphism g: U → A such that: f(-) = gL (h(-)).
This is the universal property expressing that the functor sending X to its universal enveloping algebra is left adjoint to the functor sending a unital associative algebra A to its Lie algebra AL.
Read more about this topic: Universal Enveloping Algebra
Famous quotes containing the words universal and/or property:
“For universal love is as special an aspect as carnal love or any of the other kinds: all forms of mental and spiritual activity must be practiced and encouraged equally if the whole affair is to prosper. There is no cutting corners where the life of the soul is concerned....”
—John Ashbery (b. 1927)
“All over this land women have no political existence. Laws pass over our heads that we can not unmake. Our property is taken from us without our consent. The babes we bear in anguish and carry in our arms are not ours.”
—Lucy Stone (1818–1893)