Virtual Extension of Temperley–lieb Algebra

The virtual knot theory is a new interesting subject in the recent study of low dimensional topology. In this paper, we explore the algebraic structure underlying the virtual braid group and call it the virtual Temperley–Lieb algebra which is an extension of the Temperley–Lieb algebra by adding the group algebra of the symmetrical group. We make a connection clear between the Brauer algebra and virtual Temperley–Lieb algebra, and show the algebra generated by permutation and its partial transpose to be an example for the virtual Temperley–Lieb algebra and its important quotients.