Toroidal Lie algebras and Bogoyavlensky's (2 + 1)-dimensional equation

We introduce an extension of the ℓ-reduced KP hierarchy, which we call the ℓ-Bogoyavlensky hierarchy. Bogoyavlensky's 2 + 1-dimensional extension of the KdV equation is the lowest equation of the hierarchy in case of ℓ = 2. We present a group-theoretic characterization of this hierarchy on the basis of the 2-toroidal Lie algebra $\textbf{s}\mathbf\ell_\mathbf\ell^\textbf{tor}$. This reproduces essentially the same Hirota bilinear equations as those recently introduced by Billig and Iohara et al. We can further derive these Hirota bilinear equations from a Lax formalism of the hierarchy. This Lax formalism also enables us to construct a family of special solutions that generalize the breaking soliton solutions of Bogoyavlensky. These solutions contain the N-soliton solutions, which are usually constructed by use of vertex operators.