LEADER 03616nam 2200481 450 001 9910793296103321 005 20220528001309.0 010 $a1-4704-4823-8 035 $a(CKB)4100000007133852 035 $a(MiAaPQ)EBC5571105 035 $a(Au-PeEL)EBL5571105 035 $a(OCoLC)1065248741 035 $a(RPAM)20649665 035 $a(PPN)231946414 035 $a(EXLCZ)994100000007133852 100 $a20220528d2018 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aCluster algebras and triangulated surfaces$hPart II$iLambda lenghts /$fSergey Fomin, Dylan Thurston 210 1$aProvidence, Rhode Island :$cAmerican Mathematical Society,$d[2018] 210 4$dİ2018 215 $a1 online resource (110 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vVolume 255, number 1223 311 $a1-4704-2967-5 320 $aIncludes bibliographical references. 327 $aCover -- Title page -- Chapter 1. Introduction -- Chapter 2. Non-normalized cluster algebras -- Chapter 3. Rescaling and normalization -- Chapter 4. Cluster algebras of geometric type and their positive realizations -- Chapter 5. Bordered surfaces, arc complexes, and tagged arcs -- Chapter 6. Structural results -- Chapter 7. Lambda lengths on bordered surfaces with punctures -- Chapter 8. Lambda lengths of tagged arcs -- Chapter 9. Opened surfaces -- Chapter 10. Lambda lengths on opened surfaces -- Chapter 11. Non-normalized exchange patterns from surfaces -- Chapter 12. Laminations and shear coordinates -- Chapter 13. Shear coordinates with respect to tagged triangulations -- Chapter 14. Tropical lambda lengths -- Chapter 15. Laminated Teichmu?ller spaces -- Chapter 16. Topological realizations of some coordinate rings -- Chapter 17. Principal and universal coefficients -- Appendix A. Tropical degeneration and relative lambda lengths -- Appendix B. Versions of Teichmu?ller spaces and coordinates -- Bibliography -- Back Cover. 330 $aFor any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, the authors construct a geometric realization in terms of suitable decorated Teichmu?ller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. The authors' model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from the authors' previous paper, removing unnecessary assumptions on the surface. 410 0$aMemoirs of the American Mathematical Society ;$vVolume 255, number 1223. 606 $aCluster algebras 615 0$aCluster algebras. 676 $a512.44 700 $aFomin$b Sergey$01544047 702 $aThurston$b Dylan P.$f1972- 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910793296103321 996 $aCluster algebras and triangulated surfaces$93797931 997 $aUNINA