Toroidal Lie algebras and Bogoyavlensky ’ s 2 + 1 - dimensional equation

We introduce an extension of the l-reduced KP hierarchy, which we call the lBogoyavlensky 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 sl l . 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 equation 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.