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I have a problem understanding when an algebra is closed.

For example the angular momentum algebra closes:

$$[L_i,L_j]=\epsilon_{ijk}L_k$$

but would the algebra still be closed if, for example, there would be a change like this:

$$[L_1,L_2]=L_3 + a$$

all the other brackets unchanged?

mattiav27
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    As far as I can tell, if something is not "closed" the way you seem to mean it (for $x,y \in L$, we have $[x,y] \in L$), then it's not an algebra. – Torsten Schoeneberg Jan 14 '21 at 07:43

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So two answers here.

Yes it would still be closed if you do one of two things: redefine the other Lie brackets to contain $L_3 +a$ instead of $L_3$ OR if you add $1$ to your Lie algebra so that you can break the sum apart. Technically $[L_3+a,L_1] = [L_3, L_1] + [a, L_1]$ requires that both $L_3$ and $1$ be in your Lie algebra. $1$ is not a standard element of the angular momentum Lie algebra ($\mathfrak{su}(2)$). Of course in a quantum mechanical world where these are operators, the Lie algebra actually exists inside of a larger structure, so maybe by borrowing from that, you can get away with not doing the above and the closure exists a fortiori. (Note that adding $L_3+a$ to your Lie algebra is the same as adding $1$ since $L_3+a-L_3 = a$.)

The other answer is: provided you have an actual Lie bracket, you can always close your Lie algebra when given a generating set. You just might not like what you end up with.. For instance, a Lie algebra closely related to yours which appears in mathematical physics is what I call a Smith type (based on the person that originally heavily studied them). It's generated by $[H, L] = -L$, $[H, R] = +R$, and $[L, R] = H^2$. At first glance this doesn't seem terrible, but using a certain trick (which I independently rediscovered almost 30 years later (no I'm not bummed out about it..)) that Smith employs, you can show that this Lie algebra, despite being finitely generated, is not finite dimensional. In other words, you have three base relations but from those you can create an infinite number of linearly independent elements in the algebra.

Cameron Williams
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