HDB1 said:
Please, I have a question about universal enveloping algebra: Let ##U=U(\mathfrak{g})## be the quotient of the free associative algebra ##\mathcal{F}## with generators ##\left\{a_i: i \in I\right\}## by the ideal ##\mathcal{I}## generated by all elements of the form ##a_i a_j-a_j a_i-\sum_{k \in I} c_{i, j}^k a_k##. The associative algebra ##U(\mathfrak{g}):=\mathcal{F} / \mathcal{I}## is called the universal enveloping algebra.
1- Please, could I know what is free algebra here, and why universal enveloping algebra is associative?
2- and please, if you could explain to me the universal enveloping algebra of lie algebra ##\mathfrak{s l}_2##,Thanks in advance,
I hate the universal enveloping algebra. It is sometimes necessary but in general of little help. It is associative because we construct it to be associative. We gather all we need as elements, which are all elements of the Lie algebra, make an associative algebra out of it, and in order that it still has anything to do with our original Lie algebra, we identify the multiplication of the Lie algebra as if it came from ordinary matrix multiplication.
Example: ##\mathfrak{g}=\mathfrak{sl}(2).##
This means we need primarily the elements ##\{E,H,F\}.## But now, consider them as just letters; an alphabet with ##3## letters if you like. Our free algebra now will consist of all linear combinations of any finite words over that alphabet. E.g.
$$
FHE \in \mathcal{F}\; , \; H^3EH\in \mathcal{F}\; , \;FEHEFHEHEFHEF\in \mathcal{F}\; , \;E^2\in \mathcal{F}
$$
We require associativity per construction. And we have the linear span of these words. That thing is huge!
In the second step, we consider the elements ##HE-EH-2E\, , \,HF-FH+2F\, , \,EF-FE-H## which are already in ##\mathcal{F}.## They span a subspace and generate an ideal in ##\mathcal{I}\subseteq \mathcal{F}.## And I have no imagination of what it looks like. The good news is, we do not have to care. We are interested in the quotient (associative) algebra ##U(\mathfrak{g})=\mathcal{F}/\mathcal{I}.## This means the ideal becomes our new zero. But what does it mean if the elements of ##\mathcal{I}## are considered to be zero? It means that
\begin{align*}
HE-EH-2E=0 &\Longleftrightarrow HE-EH=2E\\
HF-FH+2F=0 &\Longleftrightarrow HF-FH=-2F\\
EF-FE-H=0&\Longleftrightarrow EF-FE=H
\end{align*}
Now that is the reason for all the trouble: we have equations in an
associative algebra that pretend to be Lie multiplications of matrices. Remember that ##[H,E]=HE-EH=2E\, , \,[H,F]=HF-FH=-2F\, , \,[E,F]=EF-FE=H## as ##2\times 2## matrices with vanishing trace. However, we constructed something
like an associative hull of our Lie algebra, therefore the name
enveloping. And
universal because we put in as many elements as ever possible. Ok, the correct mathematical reason why it is called
universal is a bit more sophisticated than that, but not of interest here. Here is the full dose (to deter):
https://ncatlab.org/nlab/show/universal+enveloping+algebra
1.) The equations we get from the quotient by ##\mathcal{I}## allow replacing ##HE## by ##2E+EH## so we can sort ##H## and ##E## and similar for ##H,F## and ##E,F.## Hence the words in ##\mathcal{F}## can be written as linear combinations of ##E^pH^qF^r## with ##r,p,q\in \mathbb{N}_0## which are thus the typical elements in ##U(\mathfrak{g}).## However, this might be different in the cases where the multiplications in ##\mathfrak{g}## are more complicated.
2.) Just a remark about
quotient (Lie) algebras. Say we consider the example ## \mathbb{Z} / 3\mathbb{Z} ## again. Some authors say
factor (Lie) algebras instead of 'quotient' and that ##3\mathbb{Z}## is
factored out. Both are crutches since it is neither a quotient nor a factor. However, ##\mathbb{Z}/3\mathbb{Z}## is a partition of ##\mathbb{Z}## into the three sets ##0+3\mathbb{Z}\, , \,1+3\mathbb{Z}\, , \,2+3\mathbb{Z}## which become the new elements. Hence, we have built a quotient along ##3\mathbb{Z}## but we have also factored ##\mathbb{Z}## into the remainder classes ##0,1,2## which we get from a division by ##3##. So whenever someone says quotient space or factor space, then a construction ##A/B## is meant.
Don't spend too much thought on it. But keep the construction principle in mind, it occurs on many more occasions, e.g. exterior product spaces (Graßmann algebras; wedge products).