Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras IV

By Seok-Jin Kang, Masaki Kashiwara, Myungho Kim, Se-jin Oh. 2013

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Let $$U'_q(\mathfrak {g})$$Uq′(g) be a twisted affine quantum group of type $$A_{N}^{(2)}$$AN(2) or $$D_{N}^{(2)}$$DN(2) and let $$\mathfrak {g}_{0}$$g0 be the finite-dimensional simple Lie algebra of type $$A_{N}$$AN or $$D_{N}$$DN. For a Dynkin quiver of type $$\mathfrak {g}_{0}$$g0, we define a full subcategory $${\mathcal C}_{Q}^{(2)}$$CQ(2) of the category of finite-dimensional integrable $$U'_q(\mathfrak {g})$$Uq′(g)-modules, a twisted version of the category $${\mathcal C}^{(1)}_{Q}$$CQ(1) introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur–Weyl duality, we construct an exact faithful KLR-type duality functor $${\mathcal F}_{Q}^{(2)}:\mathrm{Rep}(R) \rightarrow {\mathcal C}_{Q}^{(2)}$$FQ(2):Rep(R)→CQ(2), where $$\mathrm{Rep}(R)$$Rep(R) is the category of finite-dimensional modules over the quiver Hecke algebra R of type $$\mathfrak {g}_{0}$$g0 with nilpotent actions of the generators $$x_k$$xk. We show that $${\mathcal F}_{Q}^{(2)}$$FQ(2) sends any simple object to a simple object and induces a ring isomorphism .

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e6e38aa7785070fc033ceaaba67899d3ea2d781a