Speaker
Description
Let $\mathfrak{g}$ be an affine Lie algebra and $Y_{\hbar}(\mathfrak{g})$ be the Yangian associated to $\mathfrak{g}$. Unlike its finite counterpart, the affine Yangian is not known to possess a universal $R$--matrix. In particular, one does not immediately have solutions of the quantum Yang--Baxter equation on an appropriate category of representations of the affine Yangian. The sole exception is the Maulik--Okounkov theory, which provides rational solutions to QYBE, on representations coming from the geometry of quiver varieties.
In this talk I will present a construction of two meromorphic $R$--matrices, related by a unitarity relation, for category $\mathcal{O}$ representations of $Y_{\hbar}(\mathfrak{g})$. I will show that our $R$--matrices can be normalized on highest--weight representations in order to obtain rational solutions to QYBE. This talk is based on joint works with Andrea Appel, Valerio Toledano Laredo and Curtis Wendlandt.
