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

International audienceThe Yang-Baxter σ-model is an integrable deformation of the principal chiral model on a Lie group G. The deformation breaks the G × G symmetry to U(1)$^{rank(}^{G}^{)}$ × G. It is known that there exist non-local conserved charges which, together with the unbroken U(1)$^{rank(}^{G}^{)}$ local charges, form a Poisson algebra , which is the semiclassical limit of the quantum group $ {U}_q\left(\mathfrak{g}\right) $ , 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 , the classical analogue of the quantum loop algebra $ {U}_q\left(L\mathfrak{g}\right) $ , 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 σ-model