Projectors in the virtual Temperley–Lieb algebra

In this paper, 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 [Formula: see text] in terms of its axiomatic properties in the virtual Temperley–Lieba algebra [Formula: see text]. Finally, we find the coefficients of [Formula: see text] and give an explicit formula for the projector [Formula: see text].