Abstract
We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q, t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q, t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defined in a previous work as a quantum cluster algebra, contains indeed the well-known quantum Grothendieck ring of the category of finite-dimensional representations. Finally, we display our algorithm on a concrete example.
Highlights
Let g be a finite-dimensional simple Lie algebra, and Uq (g) be the quantum affine algebra
Chari–Pressley classification result implies that every simple module can be obtained as a subquotient of a tensor product of such fundamental modules
In this article we propose to show that, when g is of -laced type, the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional Uq (g)-modules has a quantum cluster algebra structure (Proposition 7.3.3)
Summary
Let us fix some notations for the rest of the paper. Let g be a simple Lie algebra of rank n and of type A, D or E. This restriction is necessary as one of the main arguments of the proof is the quantum T -systems, which have only been proven for these types as yet. Let γ be the Dynkin diagram of g and let I := {1, . ), Let us denote by (αi )i∈I the simple fundamental weights. The Dynkin diagram of g is numbered as in [28], and let a1, a2, . Let h be the (dual) Coxeter number of g:
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
