We present a method of defining projectors in the virtual Temperley-Lieb algebra, that generalizes the Jones-Wenzl projectors in Temperley-Lieb algebra. We show that the projectors have similar properties with the Jones-Wenzl projectors, and contain an extra property which is associated with the virtual generator elements, that is, the product of a projector with a virtual generator is unchanged. We also show the uniqueness of the projector fₙ in terms of its axiomatic properties in the virtual Temperley-Lieba algebra VTLₙ(d). Finally, we find the coefficients of fₙ and give an explicit formula for the projector fₙ.
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Deng et al. (2025) studied this question.
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