Let $U(L)$ and $U(A)$ be the universal enveloping algebras of Lie algebra $L$ and its given subalgebra $A$. We consider $U(L)/U(A)$. It is clear that $U(L)/U(A)$ is a left $U(A)$-module. But I want to prove that it is free left module. May you please let me know the way of showing that?
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1I think this follows from Poincaré-Birkhoff-Witt. Use an ordered basis of $L$ that begins with a basis of $A$. – Jyrki Lahtonen Jul 04 '16 at 14:26
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We should construct basis of $U(L)$ under the name $Y$ consisting of basis $U(A)$ , and then define a subset of $Y$ such that the elements of this subset are not in the basis of $U(A)$ then we are be able to find for instance a basis $y_{i_{1}}+ U(A),...,y_{i_{k}}+U(A)$ for $U(L)/U(A)$ ? am I right? – Nil Jul 04 '16 at 17:07