Vertex Representations For Toroidal Lie Algebra Of Type G₂

In [MRY],[RM], the authors gave the vertex representations of the simply-laced toroidal Lie algebra, and according to the above, [T2] gave the construction of the vertex reprsentation of toroidal Lie algebra B_l. According to the main idea of [T2], we give the constructon of the vertex representation of the toroidal Lie algebra T(G₂). This construction has the close relation with the Dynkin diagram of type D₄⁽¹⁾ and a 2-cocycle.Similarly to [FLM], [MRY], we define an intergal lattice Q which contains two affine root lattices Q(D₄⁽¹⁾) and Q(G₂⁽¹⁾) of affine Lie algebras D₄⁽¹⁾ ,G₂⁽¹⁾, and define a 2-cocycle map: ε : Q × Q → {k\k⁶ = 1}. As usual, we can form a group algebra C[Q] on this lattice, and the twisted product is: e^βe^γ = ε(β,γ)e^β+γ,(?)β,γ ∈ Q. And then, we define the Fock space: V := C[Q] (?) S((H|^)₀^-), and give the construction of the vertex operators of T(G₂) and the main result (Theroem 3.3). Finally, as the vertex operators of type D₄ satisfy the equations (2.3)-(2.5), we prove the main result by using Lemma 2.3.