Riordan trees and the homotopy sl₂ weight system

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• 作者：Jean-Baptiste Meilhan^a ; ^{jean-baptiste.meilhan@ujf-grenoble.fr" class="auth_mail" title="E-mail the corresponding author} ; Sakie Suzuki^b ; ^{sakie@kurims.kyoto-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：57M27 ; 17B37
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：221
• 期：3
• 页码：691-706
• 全文大小：517 K

文摘

The purpose of this paper is twofold. On one hand, we introduce a modification of the dual canonical basis for invariant tensors of the 3-dimensional irreducible representation of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916301098&_mathId=si1.gif&_user=111111111&_pii=S0022404916301098&_rdoc=1&_issn=00224049&md5=96a1c06dc9514427f8365f7dd77141fd" title="Click to view the MathML source">U_q(sl₂)class="mathContainer hidden">class="mathCode">$U_q(s{l}_2)$, given in terms of Jacobi diagrams, a central tool in quantum topology. On the other hand, we use this modified basis to study the so-called homotopy class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916301098&_mathId=si2.gif&_user=111111111&_pii=S0022404916301098&_rdoc=1&_issn=00224049&md5=b4ccc4b4bd89470074a3d5fcf6f0a6f9" title="Click to view the MathML source">sl₂class="mathContainer hidden">class="mathCode">$s{l}_2$ weight system, which is its restriction to the space of Jacobi diagrams labeled by distinct integers. Noting that the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916301098&_mathId=si2.gif&_user=111111111&_pii=S0022404916301098&_rdoc=1&_issn=00224049&md5=b4ccc4b4bd89470074a3d5fcf6f0a6f9" title="Click to view the MathML source">sl₂class="mathContainer hidden">class="mathCode">$s{l}_2$ weight system is completely determined by its values on trees, we compute the image of the homotopy part on connected trees in all degrees; the kernel of this map is also discussed.