We introduce a sequence of q-characters of standard modules of a quantum affine algebra and we prove it has a limit as a formal power series. For g "ŝl 2 , we establish an explicit formula for the limit which enables us to construct corresponding asymptotical standard modules associated to each simple module in the category O of a Borel subalgebra of the quantum affine algebra. Finally, we prove a decomposition formula for the limit formula into q-characters of simple modules in this category O. Contents 1 Background 4 2 Limits of q-characters of standard modules 12 3 Asymptotical standard modules 16 4 Decomposition of the q-character of asymptotical standard modules 27 pN q J 5)Then, using (3.4), one has, for m ě 0,as U q pbqX`acts by 0 on u pN q , which is a highest ℓ-weight vector and v pi 0 q J PS i 0 , which is a U q pbq-module. That way the action of pxmq mě0 on v J is defined asis a linear combination of v pi 0 q K , with maxpKq ď i 0 . Hence, xm¨v J is the same linear combination, but with the v K instead of the v pi 0 q K .Action of U q pbq 0 : The algebra U q pbq 0 is generated by the ph r , k˘1 1 q rě1 .As we normalized the action of k 1 , for all N ą i 0 , k 1¨vFor the action of the h r 's, let us write, for N ą i 0 , v pN q J as in (3.5). Then, using (3.3), one has, for r ě 1, h r¨v pN q J "´h r¨v pi 0 q J¯b u pN q`v pi 0 q J b´h r¨u pN q¯`0
We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category of representations of the quantum loop algebra introduced by Hernandez–Jimbo. We use the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type , we prove that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite‐dimensional representations of the associated quantum affine algebra. In type , we identify remarkable relations in this quantum Grothendieck ring.
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.
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 pq, tq-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these pq, tq-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.
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