Bosonic realizations of higher-level toroidal Lie algebras
Toroidal Lie algebras are natural generalizations of the affine Kac-Moody Lie algebras, where the ring of Laurant polynomials in one variable (affine case) is now replaced with the ring of Laurent polynomials in several variables (toroidal case). The variable should be viewed as the coordinates of the torus, which explains the name ``toroidal''. It is known that every affine Lie algebra extends to a semidirect product with the \((c=0)\) Virasoro algebra. Similarly, a toroidal algebra admits a central extension with the space of Kähler differentials modulo exact terms. In the paper under review the authors construct a bosonic representation of the 2-toroidal algebra associated with the Lie algebra \(sl_2\), with rather mild constraints on the level. It is fair to say that this construction is a toroidal counterpart of the Wakimoto's realization for \(\hat{sl}_2\). The main results are Theorem 3.5 (the actual realization), Theorem 4.2 (module structure) and Theorem 4.3 (explicit computation of the characters).
scientific article; zbMATH DE number 5011485