# The use of Schur's lemma for Lie algebras in physics (CFT)

Anytime a one-dimensional central extension appears in the physics literature, immediately they assume that in any irreducible representation the central charge will be a multiple of the identity, implicitly (and sometimes explicitly) using Schur's Lemma (for Lie algebras). However, the version of the Schur's Lemma which tells us this is only valid for finite-dimensional modules, and physicists are usually interested in representing Hilbert spaces which in general can have infinite dimension.

Is it somehow still justified to say that the central charge is a multiple of the identity even for infinite-dimensional irreducible representations?

The two examples I have in mind are the Virasoro algebra as one dimensional central extension of the Witt algebra and the universal central extension of a loop algebra, both appearing in CFT texts, and in both cases they use Schur's Lemma as described.

• I’m pretty sure Schur’s Lemma doesn’t assume finite dimensionality. Either way, if you’re working with a reductive group over the complex numbers then you only have finite dimensional irreducible representations – leibnewtz Aug 3 at 12:15
• Where is the question? – Steven Landsburg Aug 3 at 12:47
• @StevenLandsburg I added it explicitly. – Soap Aug 3 at 14:56
• @leibnewtz It depends where you read it. That's why I said "that version", which says that for finite dimensional irreducible modules one has that elements of the center are represented as multiples of the identity. You can see this as a corollary of the Schur's lemma you are thinking about. I am wondering if this corollary somehow extends to infinite-dimensional modules. – Soap Aug 3 at 15:01
• First, there certainly do exist infinite-dimensional irreducible unitary repns of simple Lie groups, e.g., $SL_2(\mathbb R)$. Second, at least a weak version of a spectral theorem is needed to prove the corresponding Schur's lemma. – paul garrett Aug 3 at 17:46

Let $$\mathfrak{g}$$ be a complex Lie algebra with a distinguished nonzero central element $$x$$, and let $$V$$ be an irreducible representation of $$\mathfrak{g}$$. The usual proof of Schur's lemma can be adapted to show that if $$x$$ admits an eigenvector in $$V$$, then $$x$$ acts by a scalar: If $$v$$ is an eigenvector with eigenvalue $$\lambda$$, then $$v$$ lies in a submodule, namely the kernel of the central transformation $$x - \lambda$$, so irreducibility implies this kernel is equal to $$V$$.