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Monoidal categorification and quantum affine algebras

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 نشر من قبل Masaki Kashiwara
 تاريخ النشر 2019
  مجال البحث
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We introduce and investigate new invariants on the pair of modules $M$ and $N$ over quantum affine algebras $U_q(mathfrak{g})$ by analyzing their associated R-matrices. From new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable $U_q(mathfrak{g})$-modules to become a monoidal categorification of a cluster algebra.


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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 genera lize 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.
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