The affine Yangian of revisited

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• 作者：Alexander Tsymbaliuk ^{otsymbaliuk@scgp.stonybrook.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Affine Yangian ; Yangian of gl1gl1 ; Toroidal of gl1gl1 ; Representation theory ; Shuffle algebra
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：2 January 2017
• 年：2017
• 卷：304
• 期：Complete
• 页码：583-645
• 全文大小：931 K

文摘

The affine Yangian of d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311276&_mathId=si1.gif&_user=111111111&_pii=S0001870816311276&_rdoc=1&_issn=00018708&md5=6073ef4df9a3edd6ded3328df5bf2085" title="Click to view the MathML source">gl₁dden">de">${\mathfrak{gl}}_1$ has recently appeared simultaneously in the work of Maulik–Okounkov d="bbr0120">[11] and Schiffmann–Vasserot d="bbr0200">[20] in connection with the Alday–Gaiotto–Tachikawa conjecture. While the presentation from d="bbr0120">[11] is purely geometric, the algebraic presentation in d="bbr0200">[20] is quite involved. In this article, we provide a simple loop realization of this algebra which can be viewed as an “additivization” of the quantum toroidal algebra of d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311276&_mathId=si1.gif&_user=111111111&_pii=S0001870816311276&_rdoc=1&_issn=00018708&md5=6073ef4df9a3edd6ded3328df5bf2085" title="Click to view the MathML source">gl₁dden">de">${\mathfrak{gl}}_1$ in the same way as the Yangian d="mmlsi2" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311276&_mathId=si2.gif&_user=111111111&_pii=S0001870816311276&_rdoc=1&_issn=00018708&md5=32cd5d9400acbceb329dae6731833f3f" title="Click to view the MathML source">Y_h(g)dden">de">$Y_h(\mathfrak{g})$ is an “additivization” of the quantum loop algebra d="mmlsi3" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311276&_mathId=si3.gif&_user=111111111&_pii=S0001870816311276&_rdoc=1&_issn=00018708&md5=1ad010fc49747f71ec913004894b93d6" title="Click to view the MathML source">U_q(Lg)dden">de">$U_q(L\mathfrak{g})$ for a simple Lie algebra d="mmlsi4" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311276&_mathId=si4.gif&_user=111111111&_pii=S0001870816311276&_rdoc=1&_issn=00018708&md5=93cfeaa60714a61ad211e313fd875e0c" title="Click to view the MathML source">gdden">de">$\mathfrak{g}$. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311276&_mathId=si1.gif&_user=111111111&_pii=S0001870816311276&_rdoc=1&_issn=00018708&md5=6073ef4df9a3edd6ded3328df5bf2085" title="Click to view the MathML source">gl₁dden">de">${\mathfrak{gl}}_1$ by generalizing the main result of d="bbr0100">[10] to the current settings.