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We plan to introduce the Virasoro algebra and its role as a central extension and its (heuristic) emergence as infinitesimal conformal transformations of the Euclidean plane (after quantization). The main questions are as follows:

What is the Virasoro algebra? (And what is a Lie algebra?)

Why is it an interesting object?

Why is it the canonical choice for a central extension related to the quantized conformal symmetry?

The main difficulty in this topic is related to the dilemma that there does not exist a complex Virasoro group, that would’ve represented a conformal symmetry group for two-dimensional conformal field theory. Hence, one has to think of the conformal symmetry in an infinitesimal form rather than a symmetry given by an honest group.