q-Pascal triangle and representations of the Braid groups and Quantum groups (Alexandre KOSYAK, Institute of Mathematics, Kyiv, Ukraine; Max-Planck Institute for Mathematics, Bonn, Germany)

19.05.2022 15:00

We study connection between representations of the Braid group $B_n$ Lie algebra $\mathfrak{sl}_{n-1}$ and the Quantum group $U_q(\mathfrak{sl}_{n-1})$. The usual Pascal triangle appears very naturally in representations of the group $B_3$. We show how these representations come from finite-dimensional irreducible representations of a Lie algebra $\mathfrak{sl}_2$. Surprisingly, the q-Pascal triangle also appears in representations of $B_3$, due to the description by Tuba and Wenzl (2001) of low dimensional representations of $B_3$. They did not notice that q-Pascal triangle appears in their formulas. The representations connected with q-Pascal triangle should clearly come from that of $U_q(\mathfrak{sl}_{n-1})$, but the connection is still not well understood.
The two-parameter Lawrence (1990), or Lawrence-Krammer (2002), representation of $B_n$ were used to prove that the Braid groups $B_n$ are linear: Krammer for $B_4$ (2002), Bigelow for all $B_n$ (2001). We show in addition that the Lawrence representation of $B_n$ is a quantization of the symmetric square of the Burau representation.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-15, Séminaire "Topologie et géométrie"

entrée libre