On the loop Hecke algebra

Abstract: To any braid group $B_n$ ​ there is an associated (Iwahori-)Hecke algebra $H_q ​ (n)$. Over time this algebra has shown to be as intriguing as $B_n$. For example, $H_q ​ (n)$ possesses a representation for which it is in a Schur–Weyl relation with $U_q ​ (sl_d). One possible interpretation of classical braid groups is as a fundamental group of the space of configurations of $n$ distinct points in $R^2$. Taking this motion group perspective, it is natural to consider configurations of $n$ unit circles $S^1$. This yields the so-called Loop Braid group. Damiani–Martin–Rowell associated an analogue of the Hecke algebra and made a conjecture on the dimension of this Loop Hecke algebra. In this talk we will firstly briefly introduce the mentioned objects and subsequently tell about how the above Schur–Weyl picture adapts to the Loop setting. In the last part of the talk we will discuss the simple representations and the Jacobson radical.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar