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:
“So universal and widely related is any transcendent moral greatness, and so nearly identical with greatness everywhere and in every age,—as a pyramid contracts the nearer you approach its apex,—that, when I look over my commonplace-book of poetry, I find that the best of it is oftenest applicable, in part or wholly, to the case of Captain Brown.”
—Henry David Thoreau (1817–1862)
“Oh, had I received the education I desired, had I been bred to the profession of the law, I might have been a useful member of society, and instead of myself and my property being taken care of, I might have been a protector of the helpless, a pleader for the poor and unfortunate.”
—Sarah M. Grimke (1792–1873)