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A note on the zeroth products of Frenkel–Jing operators

Let gˆ be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable gˆ-module is the finite-dimensional irreducible g-module, where the action of the simple Lie algebra g is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level 1 Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra Uq(slˆn+1). By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator Yi(z), we obtain an infinite-dimensional vector space ⟨Yi(z)⟩. Next, we introduce an associative algebra Uq(sln+1)z, a certain quantum analogue of the universal enveloping algebra U(sln+1), and construct some infinite-dimensional Uq(sln+1)z-modules L(λi)z corresponding to the finite-dimensional irreducible Uq(sln+1)-modules L(λi). We show that the space ⟨Yi(z)⟩ carries a structure of an Uq(sln+1)z-module and, furthermore, we prove that the Uq(sln+1)z-module ⟨Yi(z)⟩ is isomorphic to the Uq(sln+1)z-module L(λi)z.

Affine Lie algebra ; Quantum affine algebra ; Quantum vertex algebra ; Highest weight module

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