Quantum Grothendieck rings, cluster algebras and quantum affine category $O$

The Defense Jury

Résumé

The aim of this thesis is to construct and study some quantum Grothendieck ring structure for the category $\mathcal{O}$ of representations of the Borel subalgebra $\mathcal{U}_{q}(\hat{\mathfrak{b}})$ of a quantum affine algebra $\mathcal{U}_{q}(\hat{\mathfrak{g}})$. First of all, we focus on the construction of asymptotical standard modules, analogs in the context of the category $\mathcal{O}$ of the standard modules in the category of finite-dimensional $\mathcal{U}_{q}(\hat{\mathfrak{g}})$-modules. A construction of these modules is given in the case where the underlying simple Lie algebra $\mathfrak{g}$ is $\mathfrak{sl}_{2}$. Next, we define a new quantum torus, which extends the quantum torus containing the quantum Grothendieck ring of the category of finite-dimensional modules. In order to do this, we use notions linked to quantum cluster algebras. In the same spirit, we build a quantum cluster algebra structure on the quantum Grothendieck ring of a monoidal subcategory $\mathscr{C}_{\mathbb{Z}}^{-}$ of the category of finite-dimensional representations. With this quantum torus, we define the quantum Grothendieck ring $K_t(\mathcal{O}^{+}_\mathbb{Z})$ of a subcategory $\mathcal{O}^+_\mathbb{Z}$ of the category $\mathcal{O}$ as a quantum cluster algebra. Then, we prove that this quantum Grothendieck ring contains that of the category of finite-dimensional representation. This result is first shown directly in type $A$, and then in all simply-laced types using the quantum cluster algebra structure of $K_t(\mathscr{C}_\mathbb{Z}^{-})$. Finally, we define $(q,t)$-characters for some remarkable infinite-dimensional simple representations in the category $\mathcal{O}^{+}_\mathbb{Z}$. This enables us to write $t$-deformed analogs of important relations in the classical Grothendieck ring of the category $\mathcal{O}$, which are related to the corresponding quantum integrable systems.

Publication
PhD Thesis
Léa Bittmann
Léa Bittmann
Research Associate