Lie-Poisson theory for direct limit Lie algebras

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• 作者：Mark Colarusso^a ; ^{colaruss@uwm.edu" class="auth_mail" title="E-mail the corresponding author} ; Michael Lau^b ; ^{Michael.Lau@mat.ulaval.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14L30 ; 20G20 ; 37K30 ; 53D17 ; 17B65
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：220
• 期：4
• 页码：1489-1516
• 全文大小：625 K

文摘

In the first half of this paper, we develop the fundamentals of Lie–Poisson theory for direct limits 625&_mathId=si1.gif&_user=111111111&_pii=S0022404915002625&_rdoc=1&_issn=00224049&md5=4ade0fe13681947042b14ae820046966">625-si1.gif">$G=\underrightarrow{\lim }{G}_n$ of complex algebraic groups and their Lie algebras 625&_mathId=si2.gif&_user=111111111&_pii=S0022404915002625&_rdoc=1&_issn=00224049&md5=e8815d07b8385942d0f3325731d5edae">625-si2.gif">$\mathfrak{g}=\underrightarrow{\lim }{\mathfrak{g}}_n$. We describe the Poisson pro- and ind-variety structures on 625&_mathId=si21.gif&_user=111111111&_pii=S0022404915002625&_rdoc=1&_issn=00224049&md5=fe252c12900303a6887714c5685d1b07">625-si21.gif">${\mathfrak{g}}^{⁎}=\underleftarrow{\lim }{\mathfrak{g}}_n^{⁎}$ and the coadjoint orbits of G  , respectively. While the existence of symplectic foliations remains an open question for most infinite-dimensional Poisson manifolds, we show that for direct limit algebras, the coadjoint orbits give a weak symplectic foliation of the Poisson provariety 625&_mathId=si24.gif&_user=111111111&_pii=S0022404915002625&_rdoc=1&_issn=00224049&md5=ca1bb1fad64b9672965d733037d6bfeb" title="Click to view the MathML source">g^⁎${\mathfrak{g}}^{⁎}$.

The second half of the paper applies our general results to the concrete setting of 625&_mathId=si5.gif&_user=111111111&_pii=S0022404915002625&_rdoc=1&_issn=00224049&md5=17e1539f4af9b0021cf144cc8e1a75bf" title="Click to view the MathML source">G=GL(∞)$G=\mathit{GL}(\infty )$ and 625&_mathId=si6.gif&_user=111111111&_pii=S0022404915002625&_rdoc=1&_issn=00224049&md5=273b33601e92372ae33ff68780f2f586" title="Click to view the MathML source">g^⁎=M(∞)${\mathfrak{g}}^{⁎}=M(\infty )$, the space of infinite-by-infinite complex matrices with arbitrary entries. We use the Poisson structure of 625&_mathId=si24.gif&_user=111111111&_pii=S0022404915002625&_rdoc=1&_issn=00224049&md5=ca1bb1fad64b9672965d733037d6bfeb" title="Click to view the MathML source">g^⁎${\mathfrak{g}}^{⁎}$ to construct an integrable system on 625&_mathId=si7.gif&_user=111111111&_pii=S0022404915002625&_rdoc=1&_issn=00224049&md5=0c5c860f789bfe5d28458bdbfffcf0ea" title="Click to view the MathML source">M(∞)$M(\infty )$ that generalizes the Gelfand–Zeitlin system on 625&_mathId=si8.gif&_user=111111111&_pii=S0022404915002625&_rdoc=1&_issn=00224049&md5=b8e5ab9550129066043f4c02472648bf" title="Click to view the MathML source">gl(n,C)$\mathfrak{gl}(n,\mathbb{C})$ to the infinite-dimensional setting. We further show that this integrable system integrates to a global action of a direct limit group on 625&_mathId=si7.gif&_user=111111111&_pii=S0022404915002625&_rdoc=1&_issn=00224049&md5=0c5c860f789bfe5d28458bdbfffcf0ea" title="Click to view the MathML source">M(∞)$M(\infty )$, whose generic orbits are Lagrangian ind-subvarieties of the coadjoint orbits of 625&_mathId=si9.gif&_user=111111111&_pii=S0022404915002625&_rdoc=1&_issn=00224049&md5=63ea5bb2bf3089d0a860213405287ae3" title="Click to view the MathML source">GL(∞)$\mathit{GL}(\infty )$ on 625&_mathId=si7.gif&_user=111111111&_pii=S0022404915002625&_rdoc=1&_issn=00224049&md5=0c5c860f789bfe5d28458bdbfffcf0ea" title="Click to view the MathML source">M(∞)$M(\infty )$.