Modules of the toroidal Lie algebra ${\widehat{\widehat{\mathfrak{s}\mathfrak{l}}}}_2$

Highest weight modules of the double affine Lie algebra ${\widehat{\widehat{\mathfrak{s}\mathfrak{l}}}}_2$ are studied under a new triangular decomposition. Singular vectors of Verma modules are determined using a similar condition with horizontal affine Lie subalgebras, and highest weight modules are described under the condition that $c_1>0$ and $c_2=0$.