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Register a new userHigher level Fock spaces and affine Yangian
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Authors
Ryosuke Kodera
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We construct actions of the affine Yangian of type A on higher level Fock spaces by extending known actions of the Yangian of finite type A due to Uglov. This is a degenerate analog of a result by Takemura-Uglov, which constructed actions of the quantum toroidal algebra on higher level $q$-deformed Fock spaces.
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The localized equivariant homology of the quiver variety of type $A_{N-1}^{(1)}$ can be identified with the level one Fock space by assigning a normalized torus fixed point basis to certain symmetric functions, Jack($mathfrak{gl}_N$) symmetric functions introduced by Uglov. We show that this correspondence is compatible with actions of two algebras, the Yangian for $mathfrak{sl}_N$ and the affine Lie algebra $hat{mathfrak{sl}}_N$, on both sides. Consequently we obtain affine Yangian action on the Fock space.
We study braid group actions on Yangians associated with symmetrizable Kac-Moody Lie algebras. As an application, we focus on the affine Yangian of type A and use the action to prove that the image of the evaluation map contains the diagonal Heisenberg algebra inside $hat{mathfrak{gl}}_N$.
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.
For a Dynkin quiver $Q$ (of type ADE), we consider a central completion of the convolution algebra of the equivariant K-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclercs monoidal category $mathcal{C}_Q$ of modules over the quantum loop algebra $U_q(Lmathfrak{g})$ via Nakajimas homomorphism. As an application, we show that Kang-Kashiwara-Kims generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with $Q$ and Hernandez-Leclercs category $mathcal{C}_Q$, assuming the simpleness of some poles of normalized R-matrices for type E.
We show that ${rm End}_{bf U}(V_lambdaotimes V^{otimes n})$ is generated by the affine braid group $AB_n$ where ${bf U}=U_qmathfrak g(G_2)$, $V$ is its 7-dimensional irreducible representation and $V_lambda$ is an arbitrary irreducible representation.
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