This chapter introduces the nontwisted affine algebras – infinite-dimensional Lie algebras of considerable mathematical and physical interest – and searches for generalisations that preserve and enhance those special features. The affine algebras supply classic examples of Moonshine, in that the characters of their integrable modules are vectorvalued Jacobi functions for SL2(ℤ). They thread through the remainder of the book, guiding all subsequent mathematical developments. Their Lie groups are discussed in Section 3.2.6.

Algebraically, the affine algebras naturally generalise to the Kac–Moody algebras (Section 3.3.1), although that generalisation seems to lose some of their magic. In turn, the Kac–Moody algebras generalise naturally to the Borcherds–Kac–Moody algebras (Section 3.3.2), which play a significant role in Borcherds' proof of Monstrous Moonshine through their denominator identities (Section 3.4.2). Two other natural generalisations of affine algebras are described elsewhere in Section 3.3. In Section 3.4.1 we study an important special case of what we later call the orbifold construction, and in the final subsection we touch on a more recent and tangential development.

The Virasoro algebra (Section 3.1.2) plays a prominent structural role in conformal field theory (Chapter 4) and vertex operator algebras (Chapter 5); its relation to moduli spaces is a fundamental source of Moonshine itself.

Modularity from the circle

Central extensions

Let V be any (complex) vector space, and let GL(V) denote the group of all invertible linear maps V → V. A projective representation of a group G is a map P : G → GL(V) such that P(e) = I (the identity), and given any elements g, h ∈ G, there is a nonzero complex number α(g, h) such that