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I am a physicist with some background in the representation theory of finite-dimensional Lie algebras. Having now hard times with the very basics of the infinite-dimensional ones.

In most sources the infinite-dimensional Lie algebra is defined either by its generalised Cartan matrix (finite-sized) or via commutation relations between its simple roots (again, finite number). See e.g. wiki or any of the papers here.

We generate the rest of the algebra's elements by taking commutators of simple roots. For the finite-dimensional Lie algebras this process terminates at some point, and new commutators are no longer producing new (linearly independent of the ones obtained earlier) elements.

In the case of the infinite-dimensional algebras, we should be able to go infinitely far in this process.

Question 1 (see wiki for notations)

If for certain natural number $(1 - c_ij)$ we have
$$operatornamead^1-c_ij (e_i),e_j = 0 quad,$$
then how can we keep generating new elements infinitely? Looks like at some point this process should terminate, just as in the finite-dimensional case.

Question 1, rephrased

Why does relaxing the condition of positive definiteness of the Cartan matrix leads to such dramatic changes in the structure of the Lie algebra? What exactly change at the level of the commutation relations between $e_i, f_i, h_i$? (if it does change...)

Question 2

Some authors (page 13 here) write the commutation relations for the Kac-Moody and Virasoro algebras in the following way:

$$[T_m^a, T_n^b] = i f^ab_c T^c_m+n quad,$$
$$[L_m, L_n] = (m-n) L_m+n quad.$$

From these it's really obvious that one has the infinite number of linearly independent elements in the algebras. How can these form be translated to the language of the finite number of simple roots $e_i, f_i, h_i$?

Thanks.

1) Because we can build words of arbitrary length (= roots of arbitrary height) by mixing generators. For example, for the simplest affine case $A_1^(1)$ (= affine $SU(2)$ algebra), we have $ad^3(e_1)e_2 = 0$, but $ad(e_2)ad^2(e_1)e_2 neq 0$ and in fact $[ad^2(e_2)ad^2(e_1)]^n e_2 neq 0$.

2) The first set of commutation relations apply to the affine case (also known as affine Lie algebras) only, and they are missing the "level" term:
beginequation
[T_m^a, T_n^b] = if^ab_c T^c_m+n + delta_m,-n mk
endequation

In that case, you start with a finite-dimensional Lie algebra, written in terms of Chevalley relations as before (with $e_1...e_n$, etc.). You then distinguish a specific positive root $delta$ by $delta cdot alpha_i geq 0 forall i$. This root is unique.
You append to this algebra a generator $e_0$ in such a way that the generalized Cartan matrix now has determinant 0 and rank n.
The original simple roots $e_i, i=1...n$ map onto $T_0^alpha$ for $alpha$ chosen to be in the "positive" root space of the finite-dimensional algebra. The new simple root corresponds to $T^-delta_1$. Likewise the generators $f_i$ map to $T_0^-alpha$ and $T^delta_-1$. The central elements map to a subset of the $T_0^a$ in the center of the finite Lie algebra and to the "level" $k$.

The way the Virasoro algebra enters into this is as quadratic terms in the universal enveloping algebra.