# Unitary Representations in Conformal Field Theory

So I am currently studying conformal field theory from the perspective of the representation theory of Lie algebras. I am trying to understand exactly why we care about unitarizable Verma modules. For reference, I am reading "Affine Lie Algebras and Quantum Groups" and "Symmetries, Lie Algebras, and Representations". By unitarizable, I mean a module that admits a Hermitian inner product such that $(E_{\pm}^i)^\dagger=E_{\mp^i}$ and $(H^i)^\dagger=H^i$ defines an adjoint on the Chevalley-Serre basis.

From what I understand, a unitary representation $V$ of a Lie group $G$ is a representation that admits an inner product satisfying $$(Uv,Uw)=(v,w)$$ for all $v,w \in V$ and $U \in G$. This implies that $U$ is unitary. Now, if $U$ is in some connected component of the identity of $G$, we can write it in terms of elements of the corresponding Lie algbera $\mathfrak{g}$ using the exponential map. Given a basis $\{J^a\}$ for $\mathfrak{g}$, we have $$U=\text{exp}\left(\sum_{a} \alpha_a J^a \right) \ \ \ \ \ U^{-1}=\text{exp}\left(-\sum_{a} \alpha_a J^a \right).$$ Since $U$ is unitary, we can conclude that $(J^a)^\dagger=-J^a$. Extending this to complex Lie algebras gives us the required adjoint on the Chevalley-Serre basis.

Now, to my questions. First, why do we care about unitary representations of Lie groups? Second, what exactly is more fundamental here; the representation of the Lie group or the representation of the Lie algebra? I have an issue with saying that these are equivalent. For example, Lie's third theorem only applies to finite-dimensional Lie algebras (not the Virasoro algebra of affine Lie algberas, which are very fundamental in CFT). From what I understand, we are also interested in infinite-dimensional (although irreducible) representations of Lie algebras.

We care about unitary representations of Lie algebras/groups because physical symmetries corresponding to unitary operators by Wigner's theorem. So if we know that a quantum theory should have a symmetry group $G$, then we know its space of states must furnish a unitary (or rather, projective, see this Q&A of mine for the mathematical details) representation of $G$.
In particular, if the algebra includes operators that physically are related to time evolution, such as the Virasoro algebra's $L_0$ that is essentially the Hamiltonian driving time evolution, then non-unitarity of the representation means a failure of the physical probabilities to always add up to 1, i.e. the theory is inconsistent.