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We study monoidal categorifications of certain monoidal subcategories $mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional modules over quiver Hecke algebra of type A${}_infty$. In particular, when the quantum affine algebra is of type A or B, the subcategory coincides with the monoidal category $mathcal{C}_{mathfrak{g}}^0$ introduced by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.
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Let $U_q(mathfrak{g})$ be a quantum affine algebra of untwisted affine $ADE$ type, and $mathcal{C}_{mathfrak{g}}^0$ the Hernandez-Leclerc category of finite-dimensional $U_q(mathfrak{g})$-modules. For a suitable infinite sequence $widehat{w}_0= cdots s_{i_{-1}}s_{i_0}s_{i_1} cdots$ of simple reflections, we introduce subcategories $mathcal{C}_{mathfrak{g}}^{[a,b]}$ of $mathcal{C}_{mathfrak{g}}^0$ for all $a le b in mathbb{Z}sqcup{ pm infty }$. Associated with a certain chain $mathfrak{C}$ of intervals in $[a,b]$, we construct a real simple commuting family $M(mathfrak{C})$ in $mathcal{C}_{mathfrak{g}}^{[a,b]}$, which consists of Kirillov-Reshetikhin modules. The category $mathcal{C}_{mathfrak{g}}^{[a,b]}$ provides a monoidal categorification of the cluster algebra $K(mathcal{C}_{mathfrak{g}}^{[a,b]})$, whose set of initial cluster variables is $[M(mathfrak{C})]$. In particular, this result gives an affirmative answer to the monoidal categorification conjecture on $mathcal{C}_{mathfrak{g}}^-$ by Hernandez-Leclerc since it is $mathcal{C}_{mathfrak{g}}^{[-infty,0]}$, and is also applicable to $mathcal{C}_{mathfrak{g}}^0$ since it is $mathcal{C}_{mathfrak{g}}^{[-infty,infty]}$.
Let $k$ be an algebraically closed field of characteristic $0$ or $p>2$. Let $mathcal{G}$ be an affine supergroup scheme over $k$. We classify the indecomposable exact module categories over the tensor category ${rm sCoh}_{rm f}(mathcal{G})$ of (coherent sheaves of) finite dimensional $mathcal{O}(mathcal{G})$-supermodules in terms of $(mathcal{H},Psi)$-equivariant coherent sheaves on $mathcal{G}$. We deduce from it the classification of indecomposable {em geometrical} module categories over $sRep(mathcal{G})$. When $mathcal{G}$ is finite, this yields the classification of {em all} indecomposable exact module categories over the finite tensor category $sRep(mathcal{G})$. In particular, we obtain a classification of twists for the supergroup algebra $kmathcal{G}$ of a finite supergroup scheme $mathcal{G}$, and then combine it with cite[Corollary 4.1]{EG3} to classify finite dimensional triangular Hopf algebras with the Chevalley property over $k$.
In this paper, we explore a canonical connection between the algebra of $q$-difference operators $widetilde{V}_{q}$, affine Lie algebra and affine vertex algebras associated to certain subalgebra $mathcal{A}$ of the Lie algebra $mathfrak{gl}_{infty}$. We also introduce and study a category $mathcal{O}$ of $widetilde{V}_{q}$-modules. More precisely, we obtain a realization of $widetilde{V}_{q}$ as a covariant algebra of the affine Lie algebra $widehat{mathcal{A}^{*}}$, where $mathcal{A}^{*}$ is a 1-dimensional central extension of $mathcal{A}$. We prove that restricted $widetilde{V_{q}}$-modules of level $ell_{12}$ correspond to $mathbb{Z}$-equivariant $phi$-coordinated quasi-modules for the vertex algebra $V_{widetilde{mathcal{A}}}(ell_{12},0)$, where $widetilde{mathcal{A}}$ is a generalized affine Lie algebra of $mathcal{A}$. In the end, we show that objects in the category $mathcal{O}$ are restricted $widetilde{V_{q}}$-modules, and we classify simple modules in the category $mathcal{O}$.
We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of i-boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra.
Every irreducible finite-dimensional representation of the quantized enveloping algebra U_q(gl_n) can be extended to the corresponding quantum affine algebra via the evaluation homomorphism. We give in explicit form the necessary and sufficient conditions for irreducibility of tensor products of such evaluation modules.
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