The affine Yangian of revisited
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- 作者：Alexander Tsymbaliuk <sup>otsymbaliuk@scgp.stonybrook.edu" class="auth_mail" title="E-mail the corresponding authorsup>
- 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 <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" 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">gl<sub>1sub>span><span class="mathContainer hidden"><span class="mathCode"><math altimg="si1.gif" overflow="scroll"><msub><mrow><mi mathvariant="fraktur">glmi>mrow><mrow><mn>1mn>mrow>msub>math>span>span>span> has recently appeared simultaneously in the work of Maulik–Okounkov <span id="bbr0120">[11]span> and Schiffmann–Vasserot <span id="bbr0200">[20]span> in connection with the Alday–Gaiotto–Tachikawa conjecture. While the presentation from <span id="bbr0120">[11]span> is purely geometric, the algebraic presentation in <span id="bbr0200">[20]span> 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 <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" 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">gl<sub>1sub>span><span class="mathContainer hidden"><span class="mathCode"><math altimg="si1.gif" overflow="scroll"><msub><mrow><mi mathvariant="fraktur">glmi>mrow><mrow><mn>1mn>mrow>msub>math>span>span>span> in the same way as the Yangian <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" 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<sub>hsub>(g)span><span class="mathContainer hidden"><span class="mathCode"><math altimg="si2.gif" overflow="scroll"><msub><mrow><mi>Ymi>mrow><mrow><mi>hmi>mrow>msub><mo stretchy="false">(mo><mi mathvariant="fraktur">gmi><mo stretchy="false">)mo>math>span>span>span> is an “additivization” of the quantum loop algebra <span id="mmlsi3" class="mathmlsrc"><span class="formulatext stixSupport mathImg" 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<sub>qsub>(Lg)span><span class="mathContainer hidden"><span class="mathCode"><math altimg="si3.gif" overflow="scroll"><msub><mrow><mi>Umi>mrow><mrow><mi>qmi>mrow>msub><mo stretchy="false">(mo><mi>Lmi><mi mathvariant="fraktur">gmi><mo stretchy="false">)mo>math>span>span>span> for a simple Lie algebra <span id="mmlsi4" class="mathmlsrc"><span class="formulatext stixSupport mathImg" 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">gspan><span class="mathContainer hidden"><span class="mathCode"><math altimg="si4.gif" overflow="scroll"><mi mathvariant="fraktur">gmi>math>span>span>span>. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" 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">gl<sub>1sub>span><span class="mathContainer hidden"><span class="mathCode"><math altimg="si1.gif" overflow="scroll"><msub><mrow><mi mathvariant="fraktur">glmi>mrow><mrow><mn>1mn>mrow>msub>math>span>span>span> by generalizing the main result of <span id="bbr0100">[10]span> to the current settings.