This question is pretty open ended. The second part of this involves the $SL(2,{\mathbb R})$ subgroup of the Virasoro algebra. So I thought that at the risk of giving answers which might not be relevant I thought I would try to connect this with Lie theory. The Lie algebra g has a maximal set of commuting matrices that define the Cartan center $H^i$, $i~=~1,\dots,~ rank(g)$. These operators act upon the remaining operators $E^\alpha$ as $[H^i,~E^\alpha]~=~\alpha^i E^\alpha$, where $\alpha^i$ are the roots of the algebra. The Jacobi theorem $$ [[H^i,~E^\alpha],~E^\beta]~+~[[E^\beta,~H^i],~E^\alpha]~+~[[E^\alpha,~E^\beta],~H^i]~=~0 $$ permits us to compute $$ [E^\alpha,~E^\beta]~=~\matrix{ C(α,β)E^{α+β}~& :~\alpha~+~\beta~a~root \cr 2\alpha\cdot H/\alpha^2~& ~: \alpha~+~\beta~=~0\cr 0~& ~: otherwise} $$ The structure constant $|C(\alpha,\beta)~=~\pm 1$ and in the second of these the contraction of $H^i$ with the root $\alpha^i$ is a trace of $H^i$ and is used in a normalization $E^\alpha E^{-\alpha}~= 2/\alpha^2$.
The operators for the string modes obey a Virasoro algebra, $$ {[L^{a_j},~L^{b_j}]~=~(a_j~-~b_j)L^{a_j + b_j}~+~c(a_j ,b_j ).} $$ The <PERSON> generators are expanded according to the Laurent expansion $$ L^{a_n}~=~\oint\frac{dz}{2\pi iz}z^{a_n+2}T(z),~T(z)~=~-\sum_{a_n=\infty}^\infty\frac{L^{a_n}}{z^{m+2}}. $$ Commutators of the Virasoro generators $L^{-1},~L^0,~L^1$ produce the $SL(2,{\mathbb R})$ algebra $$ [L^0,~L^{-1}]~=~L_{-1},~[L^0,~L^1]~=~-L^1,~[L^1,~L^{-1}]~=~2L^0. 610
$$ This is the same in form as the $SU(2)$ algebra for the angular momentum operators $L_\pm,~L_z$, but is noncompact.
A general commutator of an element $T^a~=~T^a(z)$ in the vector space of a Lie algebra obeys $[T^a,~T^b]~=~iC^{ab}cT^c$. The inner product of these elements defines a positive element $\langle T^a,~T^b\rangle~=~h^{ab}$. This serves as a metric in the vector space of the Lie algebra. This defines a rule $$ \langle[T^a,~T^b],~T^c\rangle~+~\langle T^b,~[T^a,~T^c]\rangle~=~0. $$ So the metric $h^{ab}$ defined in some representation, $r$, of matrix element $t^a_r$ then gives the <PERSON>’s lemma result $tr(t^a_rt^b_r)~=~T_rh^{ab}$. This further gives the definition of the Coxeter number cox(g) $$ -\sum{cd}C^{ac}_dC^{bd}_c~=~cox(g)(α_L)^2h^{ab} $$ for $\alpha_L$ any long root.
With some of these Lie algebraic basics down operator produce expansions (OPE) can be found. The bosonic vertex operator for the heterotic string is of the form $j(z)\phi^i({\bar z})exp(ik\cot X)$, for $X$ the string world sheet. 364