Semi-Infinite Wedges and Vertex Operators

The level 1 highest weight modules of the quantum affine algebra Uq(ŝln) can be described as spaces of certain semi-infinite wedges. Using a q-antisymmetrization procedure, these semi-infinite wedges can be realized inside an infinite tensor product of evaluation modules. This realization gives rise to simple descriptions of vertex operators and (up to a scalar function) their compositions. 1 Representations of sl∞ 1.1 Infinite tensors and infinite wedges Let V = C be a countable-dimensional C-vector space, with basis {vi}i∈Z. The Lie algebra sl∞ = sl∞(C), consisting of infinite matrices with finitely many non-zero entries and trace 0, acts on V . The elements ei, fi, and hi, where ei · vj = δi,j−1 · vj−1 (1) fi · vj = δi,j · vj+1 (2) hi · vj = (δi,j − δi,j−1)vj, (3) generate sl∞. Consider the tensor product V ⊗ V ⊗ V ⊗ · · · . In this tensor product, sl∞ acts by ei · (vm1 ⊗ vm2 ⊗ · · ·) = ∑ j: mj=i+1 vm1 ⊗ · · · ⊗ vmj−1 ⊗ vi ⊗ vmj+1 ⊗ · · · (4) fi · (vm1 ⊗ vm2 ⊗ · · ·) = ∑ j: mj=i vm1 ⊗ · · · ⊗ vmj−1 ⊗ vi+1 ⊗ vmj+1 ⊗ · · · (5) hi · (vm1 ⊗ vm2 ⊗ · · ·) = (#{r : mr = i} −#{r : mr = i+ 1}) · vm1 ⊗ vm2 ⊗ · · · (6) To be precise, the infinite sums on the right hand side of (4)-(6) lie in an appropriate completion of V ⊗ V ⊗ V ⊗ · · · . Moreover, the action of hi given by (6) is defined only for some tensors vm1 ⊗ vm2 ⊗ · · · . One restriction that certainly suffices is that each vi should appear only finitely often among the vmj . The infinite wedge product ∧∞ V may be embedded inside V ⊗ V ⊗ V ⊗ · · · by an antisymmetrization procedure. Let S∞ denote the infinite symmetric group, which is generated