A category of modules for the full toroidal Lie algebra

Abstract

Toroidal Lie algebras are very natural multi-variable generalizations of affine Kac-Moody algebras. The theory of affine Lie algebras is rich and beautiful, having connections with diverse areas of mathematics and physics. Toroidal Lie algebras are also proving themselves to be useful for the applications. Frenkel, Jing and Wang [FJW] used representations of toroidal Lie algebras to construct a new form of the McKay correspondence. Inami et al., studied toroidal symmetry in the context of a 4-dimensional conformal field theory [IKUX], [IKU]. There are also applications of toroidal Lie algebras to soliton theory. Using representations of the toroidal algebras one can construct hierarchies of non-linear PDEs [B2], [ISW]. In particular, the toroidal extension of the Korteweg-de Vries hierarchy contains the Bogoyavlensky’s equation, which is not in the classical KdV hierarchy [IT]. One can use the vertex operator realizations to construct n-soliton solutions for the PDEs in these hierarchies. We hope that further development of the representation theory of toroidal Lie algebras will help to find new applications of this interesting class of algebras. The construction of a toroidal Lie algebra is totally parallel to the well-known construction of an (untwisted) affine Kac-Moody algebra [K1]. One starts with a finite-dimensional simple Lie algebra ġ and considers Fourier polynomial maps from an N + 1-dimensional torus into ġ. Setting tk = e ixk , we may identify the algebra of Fourier polynomials on a torus with the Laurent polynomial algebra R = C[t0 , t ± 1 , . . . , t ± N ], and the Lie algebra of the ġ-valued maps from a torus with the multi-loop algebra C[t±0 , t ± 1 , . . . , t ± N ] ⊗ ġ. When N = 0, this yields the usual loop algebra. Just as for the affine algebras, the next step is to build the universal central extension (R⊗ ġ)⊕K of R⊗ ġ. However unlike the affine case, the center K is infinite-dimensional when N ≥ 1. The infinite-dimensional center makes this Lie algebra highly degenerate. One can show, for example, that in an irreducible bounded weight module, most of the center should act trivially. To eliminate this degeneracy, we add the Lie algebra of vector fields on a torus, D = Der (R) to (R⊗ ġ)⊕K. The resulting algebra,