The nil-blob algebra: An incarnation of type $\tilde{A}_1$ Soergel calculus and of the truncated blob algebra

Abstract

We introduce a type B analogue of the nil Temperley-Lieb algebra in terms of generators and relations, that we call the (extended) nil-blob algebra. We show that this algebra is isomorphic to the endomorphism algebra of a Bott-Samelson bimodule in type Ã1. We also prove that it is isomorphic to an idempotent truncation of the classical blob algebra. Thus we provide strong evidence in favor of the recent categorical Blob vs. Soergel conjecture.