Affine q-deformed symmetry and the classical Yang-Baxter sigma-model

Abstract

The Yang-Baxter $\sigma$-model is an integrable deformation of the principal chiral model on a Lie group $G$. The deformation breaks the $G \times G$ symmetry to $U(1)^{\textrm{rank}(G)} \times G$. It is known that there exist non-local conserved charges which, together with the unbroken $U(1)^{\textrm{rank}(G)}$ local charges, form a Poisson algebra $\mathscr U_q(\mathfrak{g})$, which is the semiclassical limit of the quantum group $U_q(\mathfrak{g})$, with $\mathfrak{g}$ the Lie algebra of $G$. For a general Lie group $G$ with rank$(G)>1$, we extend the previous result by constructing local and non-local conserved charges satisfying all the defining relations of the infinite-dimensional Poisson algebra $\mathscr U_q(L \mathfrak{g})$, the classical analogue of the quantum loop algebra $U_q(L \mathfrak{g})$, where $L \mathfrak{g}$ is the loop algebra of $\mathfrak{g}$. Quite unexpectedly, these defining relations are proved without encountering any ambiguity related to the non-ultralocality of this integrable $\sigma$-model.