`2024-04-16 11:30:00``2024-04-16 12:30:00``On Lie bialgebras and derivations arising from affine Yangians``Title: On Lie bialgebras and derivations arising from affine YangiansSpeaker: Curtis Wendlandt (University of Saskatoon)Speaker's URL: https://researchers.usask.ca/curtis-wendlandt/Abstract: The theory of Yangians arose in the 1980's as an algebraic framework for systematically producing rational solutions of the celebrated quantum Yang-Baxter equation (qYBE) from theoretical physics. Roughly speaking, the mechanism by which these solutions arise is as follows: Starting from any simple Lie algebra g, one can construct a Hopf algebra, called the Yangian of g, which quantizes the Lie bialgebra g[t] of polynomials in a single variable with coefficients in g. This Hopf algebra comes equipped with a remarkable formal series R(z), called its universal R-matrix, which provides a universal, formal solution to the qYBE. The desired rational solutions are then obtained by evaluating R(z) on the tensor product of any two finite-dimensional irreducible representations of the Yangian.In recent joint work with Andrea Appel and Sachin Gautam, it has been shown that this construction admits a non-trivial generalization in which g is replaced by an affine Lie algebra. In this setting, the ordinary Yangian is upgraded to an "affine Yangian", which is an example of a quantum group of double affine type. In this talk, I will explain the semiclassical limit of this story, which entails constructing a toroidal analogue of the Lie bialgebra g[t] and studying its applications to the classical Yang-Baxter equation. Time permitting, I will highlight some unexpected consequences of this construction to the theory of affine Yangians. This is based on joint work with A. Weekes.URL associated with Seminar: https://research.math.osu.edu/reps/``MW 152``OSU ASC Drupal 8``ascwebservices@osu.edu``America/New_York``public`

`2024-04-16 11:30:00``2024-04-16 12:30:00``On Lie bialgebras and derivations arising from affine Yangians``Title: On Lie bialgebras and derivations arising from affine YangiansSpeaker: Curtis Wendlandt (University of Saskatoon)Speaker's URL: https://researchers.usask.ca/curtis-wendlandt/Abstract: The theory of Yangians arose in the 1980's as an algebraic framework for systematically producing rational solutions of the celebrated quantum Yang-Baxter equation (qYBE) from theoretical physics. Roughly speaking, the mechanism by which these solutions arise is as follows: Starting from any simple Lie algebra g, one can construct a Hopf algebra, called the Yangian of g, which quantizes the Lie bialgebra g[t] of polynomials in a single variable with coefficients in g. This Hopf algebra comes equipped with a remarkable formal series R(z), called its universal R-matrix, which provides a universal, formal solution to the qYBE. The desired rational solutions are then obtained by evaluating R(z) on the tensor product of any two finite-dimensional irreducible representations of the Yangian.In recent joint work with Andrea Appel and Sachin Gautam, it has been shown that this construction admits a non-trivial generalization in which g is replaced by an affine Lie algebra. In this setting, the ordinary Yangian is upgraded to an "affine Yangian", which is an example of a quantum group of double affine type. In this talk, I will explain the semiclassical limit of this story, which entails constructing a toroidal analogue of the Lie bialgebra g[t] and studying its applications to the classical Yang-Baxter equation. Time permitting, I will highlight some unexpected consequences of this construction to the theory of affine Yangians. This is based on joint work with A. Weekes.URL associated with Seminar: https://research.math.osu.edu/reps/ ``MW 152``Department of Mathematics``math@osu.edu``America/New_York``public`**Title: **On Lie bialgebras and derivations arising from affine Yangians**Speaker: **Curtis Wendlandt (University of Saskatoon)**Speaker's URL**: https://researchers.usask.ca/curtis-wendlandt/**Abstract: **The theory of Yangians arose in the 1980's as an algebraic framework for systematically producing rational solutions of the celebrated quantum Yang-Baxter equation (qYBE) from theoretical physics. Roughly speaking, the mechanism by which these solutions arise is as follows: Starting from any simple Lie algebra g, one can construct a Hopf algebra, called the Yangian of g, which quantizes the Lie bialgebra g[t] of polynomials in a single variable with coefficients in g. This Hopf algebra comes equipped with a remarkable formal series R(z), called its universal R-matrix, which provides a universal, formal solution to the qYBE. The desired rational solutions are then obtained by evaluating R(z) on the tensor product of any two finite-dimensional irreducible representations of the Yangian.

In recent joint work with Andrea Appel and Sachin Gautam, it has been shown that this construction admits a non-trivial generalization in which g is replaced by an affine Lie algebra. In this setting, the ordinary Yangian is upgraded to an "affine Yangian", which is an example of a quantum group of double affine type. In this talk, I will explain the semiclassical limit of this story, which entails constructing a toroidal analogue of the Lie bialgebra g[t] and studying its applications to the classical Yang-Baxter equation. Time permitting, I will highlight some unexpected consequences of this construction to the theory of affine Yangians. This is based on joint work with A. Weekes.

**URL associated with Seminar: **https://research.math.osu.edu/reps/