Derived invariants for surface algebras

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• 作者：Claire Amiot^a ; ^{claire.amiot@ujf-grenoble.fr" class="auth_mail" title="E-mail the corresponding author} ; Yvonne Grimeland^b ; ^{yvonne.grimeland@ntnu.no" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16E35 ; 16G20 ; 16G70 ; 16E10 ; 16W50
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：220
• 期：9
• 页码：3133-3155
• 全文大小：600 K

文摘

In this paper we study the derived equivalences between surface algebras, introduced by David-Roesler and Schiffler [11]. Each surface algebra arises from a cut of an ideal triangulation of an unpunctured marked Riemann surface with boundary. A cut can be regarded as a grading on the Jacobian algebra of the quiver with potential class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916000402&_mathId=si1.gif&_user=111111111&_pii=S0022404916000402&_rdoc=1&_issn=00224049&md5=26895284a78627285bba5b49345de5b8" title="Click to view the MathML source">(Q,W)class="mathContainer hidden">class="mathCode">$(Q,W)$ associated with the triangulation.

Fixing a set ϵ   of generators of the fundamental group of the surface class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916000402&_mathId=si137.gif&_user=111111111&_pii=S0022404916000402&_rdoc=1&_issn=00224049&md5=478039d847b49812d7838fdf6ea2932e" title="Click to view the MathML source">π₁(S)class="mathContainer hidden">class="mathCode">${\pi }_1(S)$, we associate to any cut d   a weight class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916000402&_mathId=si3.gif&_user=111111111&_pii=S0022404916000402&_rdoc=1&_issn=00224049&md5=67d470baec590bdb87d0ae1a2d255c51" title="Click to view the MathML source">w^ϵ(d)∈Z^2g+bclass="mathContainer hidden">class="mathCode">$w^{ϵ}(d)\in {\mathbb{Z}}^{2g+b}$, where g is the genus of S and b   the number of boundary components. The main result of the paper asserts that the derived equivalence class of the surface algebra is determined by the corresponding weight class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916000402&_mathId=si4.gif&_user=111111111&_pii=S0022404916000402&_rdoc=1&_issn=00224049&md5=df8da857ce929544d9135b26e20995d0" title="Click to view the MathML source">w^ϵ(d)class="mathContainer hidden">class="mathCode">$w^{ϵ}(d)$ up to homeomorphism of the surface. Surface algebras are gentle and of global dimension ≤2, and any surface algebras coming from the same surface class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916000402&_mathId=si17.gif&_user=111111111&_pii=S0022404916000402&_rdoc=1&_issn=00224049&md5=52407c415661c027bd3107f34aeaa81d" title="Click to view the MathML source">(S,M)class="mathContainer hidden">class="mathCode">$(S,M)$ are cluster equivalent, in the sense of [2]. To prove that the weight is a derived invariant we strongly use results about cluster equivalent algebras from [2].

Furthermore we also show that for surface algebras the invariant defined for gentle algebras by Avella-Alaminos and Geiss in [6] is determined by the weight.