# Linking formal definition of representation and field theory

As a physicist learning representation theory from a more mathematical perspective, I am having trouble initially seeing how both viewpoints fit together.

A representation $\pi$ of a Lie algebra $\mathfrak g$ is understood to be a map which is a group homomorphism,

$$\pi:\mathfrak g \to \mathfrak{gl}(V)$$

where $\mathfrak{gl}(V)$ is simply $\mathrm{End}(V)$, the group of linear operators from $V$ to itself. We then have that the linear action of the algebra $\mathfrak g$ on a vector space $V$ is given by,

$$\pi(g) V "=" gV$$

where $g\in\mathfrak g$. Thus the representation specifies how the group acts on a particular space. My question is now how we can relate this to the view of representations in quantum field theory.

As a concrete example, consider two-dimensional conformal field theory. If $|\psi\rangle$ is a primary eigenstate of both $L_0$ and $\tilde L_0$, then we can obtain a bunch of others, namely,

$$L_{-1} |\psi\rangle$$ $$L_{-1} L_{-1} |\psi \rangle, \quad L_{-2} |\psi\rangle$$ $$L_{-1}L_{-1}L_{-1} |\psi\rangle, \quad L_{-1}L_{-2}|\psi\rangle, \quad L_{-3}|\psi\rangle$$

and so forth. In the language of physics texts, it is often said we are 'building representations' of the Virasoro algebra by acting on the primary, and they are referred to as irreducible representations of the Virasoro algebra. I'd like to make this connection more precise now to representations.

1. In this example, that would $\pi$ explicitly be?
2. Presuming $\mathfrak g$ is the Virasoro algebra here, what would be the $V$ in $\pi : \mathfrak{g} \to \mathfrak{gl}(V)$ in this case?

It appears that OP essentially already knows what $\pi$ is. It seems relevant to mention that OP is describing a Verma module with a highest weight vector $|\psi \rangle$. The underlying vector space is isomorphic to $$V~=~U(\mathfrak{g}_-)\otimes_\mathbb{F}|\psi \rangle,$$where $\mathfrak{g}_-$ is the Lie subalgebra generated by the negative root spaces of $\mathfrak{g}$. Moreover, it's a fact that any (possibly irreducible) representation with same highest weight can be realized as a quotient of the Verma module.
• Are you able to explain what $V$ is in this case?
• Just to check if I have things clearly understood then. If I am given $V$ and the Lie algebra, $\mathfrak g$, since I can in theory determine a $\pi$, I can loosely say that an element of $V$ is, or offers, a representation of $\mathfrak g$, since I can recover the "real" representation $\pi$?