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On bases of quantum affine algebras

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 Added by Han Xu
 Publication date 2021
  fields
and research's language is English




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Let $textbf{U}^+$ be the positive part of the quantum group $textbf{U}$ associated with a generalized Cartan matrix. In the case of finite type, Lusztig constructed the canonical basis $textbf{B}$ of $textbf{U}^+$ via two approaches. The first one is an elementary algebraic construction via Ringel-Hall algebra realization of $textbf{U}^+$ and the second one is a geometric construction. The geometric construction of canonical basis can be generalized to the cases of all types. The generalization of the elementary algebraic construction to affine type is an important problem. We give several main results of algebraic constructions to the affine canonical basis in this ariticle. These results are given by Beck-Nakajima, Lin-Xiao-Zhang, Xiao-Xu-Zhao, respectively.



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172 - Jie Xiao , Han Xu , Minghui Zhao 2021
For quantum group of affine type, Lusztig gave an explicit construction of the affine canonical basis by simple perverse sheaves. In this paper, we construct a bar-invariant basis by using a PBW basis arising from representations of the corresponding tame quiver. We prove that this bar-invariant basis coincides with Lusztigs canonical basis and obtain a concrete bijection between the elements in theses two bases. The index set of these bases is listed orderly by modules in preprojective, regular non-homogeneous, preinjective components and irreducible characters of symmetric groups. Our results are based on the work of Lin-Xiao-Zhang and closely related with the work of Beck-Nakajima. A crucial method in our construction is a generalization of that by Deng-Du-Xiao.
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Let $Q$ be a finite acyclic valued quiver. We give the high-dimensional cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$.
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.
Let $U_q(mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $mathcal{C}_{mathfrak{g}}$ be Hernandez-Leclercs category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $mathcal{C}_{mathfrak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $Lambda$ and $Lambda^infty$ introduced by the authors. We next define the reflections $mathcal{S}_k$ and $mathcal{S}^{-1}_k$ acting on strong duality data $D$. We prove that if $D$ is a strong (resp. complete) duality datum, then $mathcal{S}_k(D)$ and $mathcal{S}_k^{-1}(D)$ are also strong (resp. complete ) duality data. We finally introduce the notion of affine cuspidal modules in $mathcal{C}_{mathfrak{g}}$ by using the duality functor $F_D$, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.
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