Quantum Yangians Research Articles

The center of the reflection equation algebra via quantum minors

We give simple formulas for the elements ck appearing in a quantum Cayley-Hamilton formula for the reflection equation algebra (RE algebra) associated to the quantum group Uq(glN), answering a question of Kolb and Stokman. The ck's are certain canonical generators of the center of the RE algebra, and hence of Uq(glN) itself; they have been described by Reshetikhin using graphical calculus, by Nazarov-Tarasov using quantum Yangians, and by Gurevich, Pyatov and Saponov using quantum Schur functions; however no explicit formulas for these elements were previously known.As byproducts, we prove a quantum Girard-Newton identity relating the ck's to the so-called quantum power traces, and we give a new presentation for the quantum group Uq(glN), as a localization of the RE algebra along certain principal minors.

AN INTRODUCTION TO YANGIAN SYMMETRIES

We review some aspects of the quantum Yangians as symmetry algebras of two-dimensional quantum field theories. The plan of these notes is the following: 1 — The classical Heisenberg model: •Non-Abelian symmetries; •The generators of the symmetries and the semi-classical Yangians; •An alternative presentation of the semi-classical Yangians; (Digression on Poisson-Lie groups. 2 — The quantum Heisenberg chain: •Non-Abelian symmetries and the quantum Yangians; •The transfer matrix and an alternative presentation of the Yangians.