# Yangians and quantum loop algebras

@article{Gautam2013YangiansAQ,
title={Yangians and quantum loop algebras},
author={Sachin Gautam and Valerio Toledano Laredo},
journal={Selecta Mathematica},
year={2013},
volume={19},
pages={271-336}
}
• Published 16 December 2010
• Mathematics
• Selecta Mathematica
Let $$\mathfrak{g }$$ be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra $$U_\hbar (L\mathfrak g )$$ of $$\mathfrak{g }$$ degenerates to the Yangian $${Y_\hbar (\mathfrak g )}$$. We strengthen this result by constructing an explicit algebra homomorphism $$\Phi$$ from $$U_\hbar (L\mathfrak g )$$ to the completion of $${Y_\hbar (\mathfrak g )}$$ with respect to its grading. We show moreover that $$\Phi$$ becomes an isomorphism when $${U_\hbar (L\mathfrak g… QUANTUM LOOP ALGEBRAS AND ℓ-ROOT OPERATORS Let g$$ \mathfrak{g} $$be a simple Lie algebra over ℂ and q ∈ ℂ× transcendental. We consider the category CP$$ {\mathcal{C}}_{\mathcal{P}} $$of finite-dimensional representations of the quantum The R-Matrix Presentation for the Yangian of a Simple Lie Algebra • C. Wendlandt • Mathematics, Physics Communications in Mathematical Physics • 2018 Starting from a finite-dimensional representation of the Yangian$${Y (\mathfrak{g})}$$Y(g) for a simple Lie algebra$$ \mathfrak{g}$$g in Drinfeld’s original presentation, we construct a Hopf Quantum Geometry and Quiver Gauge Theories • Physics, Mathematics • 2013 We study macroscopically two dimensional$${\mathcal{N}=(2,2)}$$N=(2,2) supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product Higher spins and Yangian symmetries • Physics • 2017 A bstractThe relation between the bosonic higher spin W∞λ$$ {\mathcal{W}}_{\infty}\left[\lambda \right] $$algebra, the affine Yangian of gl1$$ \mathfrak{g}{\mathfrak{l}}_1 $$, and the SHc algebra is The supersymmetric affine Yangian • Physics • 2017 A bstractThe affine Yangian of gl1 is known to be isomorphic to W1+∞$$ {\mathcal{W}}_{1+\infty } $$, the W-algebra that characterizes the bosonic higher spin — CFT duality. In this paper we propose AN EXPLICIT ISOMORPHISM BETWEEN QUANTUM AND CLASSICAL$$ \mathfrak{s}{\mathfrak{l}}_n $$• Mathematics Transformation Groups • 2019 Let $$\mathfrak{g}$$ be a complex semisimple Lie algebra. Although the quantum group $${U}_{\hslash \mathfrak{g}}$$ is known to be isomorphic, as an algebra, to the undeformed enveloping algebra Coherent states in quantum \mathcal{W}_{1+\infty} algebra and qq-character for 5d Super Yang-Mills • Physics, Mathematics • 2016 The instanton partition functions of \mathcal{N}=1 5d super Yang-Mills are built using elements of the representation theory of quantum \mathcal{W}_{1+\infty} algebra: Gaiotto state, intertwiner, The cohomological Hall algebra of a preprojective algebra • Mathematics • 2014 We introduce for each quiver Q and each algebraic oriented cohomology theory A, the cohomological Hall algebra (CoHA) of Q, as the A-homology of the moduli of representations of the Quantum N-toroidal algebras and extended quantized GIM algebras of N-fold affinization • Mathematics • 2019 We introduce the notion of quantum N-toroidal algebras uniformly, which is a natural generalization of the usual quantum toroidal algebras as well as extended quantized GIM algebras of N-fold Representations of quantum affine superalgebras We study the quantum affine superalgebra$$U_q(\mathcal {L}\mathfrak {sl}(M,N))$$Uq(Lsl(M,N)) and its finite-dimensional representations. We prove a triangular decomposition and establish a system of ## References SHOWING 1-10 OF 42 REFERENCES Meromorphic Kazhdan-Lusztig equivalence for Yangians and quantum loop algebras • Mathematics • 2014 Let {\mathfrak g} be a complex semisimple Lie algebra, and Y_h({\mathfrak g}), U_q(L{\mathfrak g}) the corresponding Yangian and quantum loop algebra, with deformation parameters related by Isomorphism of two realizations of quantum affine algebra$$U_q (\widehat{\mathfrak{g}\mathfrak{l}{\text{(}}n{\text{)}}})$$• Mathematics • 1993 AbstractWe establish an explicit isomorphism between two realizations of the quantum affine algebra$$U_q (\widehat{\mathfrak{g}\mathfrak{l}{\text{(}}n{\text{)}}})$$given previously by Drinfeld and Quantum affine algebras • Mathematics • 1991 AbstractWe classify the finite-dimensional irreducible representations of the quantum affine algebra$$U_q (\hat sl_2 ) in terms of highest weights (this result has a straightforward
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