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Название: Operads in Algebra, Topology and Physics
Авторы: Markl M., Shnider S., Stasheff J.
Аннотация:
Operads are mathematical devices which describe algebraic structures of many varieties and in various categories. Operads are particularly important in categories with a good notion of "homotopy" where they play a key role in organizing hierarchies of higher homotopies. Significant examples first appeared in the sixties though the formal definition and appropriate generality waited for the seventies. These early occurrences were in algebraic topology in the study of (iterated) loop spaces and their chain algebras. In the nineties, there was a renaissance and further development of the theory inspired by the discovery of new relationships with graph cohomology, representation theory, algebraic geometry, derived categories, Morse theory, symplectic and contact geometry, combinatorics, knot theory, moduli spaces, cyclic cohomology, and, not least, theoretical physics, especially string field theory and deformation quantization. The generalization of quadratic duality (e.g., Lie algebras as dual to commutative algebras) together with the property of Koszulness in an essentially operadic context provided an additional computational tool for studying homotopy properties outside of the topological setting. The book contains a detailed and comprehensive historical introduction describing the development of operad theory from the initial period when it was a rather specialized tool in homotopy theory to the present when operads have a wide range of applications in algebra, topology, and mathematical physics. Many results and applications currently scattered in the literature are brought together here along with new results and insights. The basic definitions and constructions are carefully explained and include many details not found in any of the standard literature. There is a chapter on topology, reviewing classical results with the emphasis on the $W$-construction and homotopy invariance. Another chapter describes the (co)homology of operad algebras, minimal models, and homotopy algebras. A chapter on geometry focuses on the configuration spaces and their compactifications. A final chapter deals with cyclic and modular operads and applications to graph complexes and moduli spaces of surfaces of arbitrary genus.
modular n-crolla of genus g271 n-star250 -algebra195 -algebra13172195 -category14 -map113 -space10 Baues operad30 -algebra30 Hochschild cochains of A with coefficients in itself29104 Hochschild chains of A with coefficients in itself259 category of K-collections85 configuration space of n distinct labeled points in 26100 coendomorphism operad43 -algebra cochain complex with coefficients in itself177339 -algebra chain complex with trivial coefficients173 -algebra19172196 category of differential graded pseudo-operads126 -module of decomposables of an operad 187 , standard unit k-dimensional disk (ball)204 set of edges of a graph 269 linear dual of a -module141 Czech dual of a -module E142 operad for Gerstenhaber algebras30176 -operad12 set of flags (half-edges) of a graph 269 simplified notation for the operad 27226 simplified notation for the space 27225 operad 225 compactification of the moduli space 225 Feynman transform of a -modular operad A287 genus of a labeled graph 270 graph complex301 set of connected graphs with Euler characteristic 1-n301 set of graphs with a metric301 minimal model of the operad 30 Harrison homology of A with coefficients in M266 Hochschild homology of A with coefficients in itself259 de Rham cohomology of an algebra A260 operadic cohomology of a -algebra V with coefficients in itself177 cyclic homology of an associative algebra A259 -space104 set of internal flags of a graph 285 category of isomorphisms of a category 72271 Stasheff's associahedron956 -algebra172195 set of legs (half-edges) of a graph 269 -algebra17195 mapping class group of a genus g surface with s distinct labeled points292 permutahedron97 -module of indecomposables of an operad 188 ribbon graph complex295 set of ribbon graphs with a metric292 set of ribbon graphs such that has s holes and genus g292 category of finite sets40 category of -modules40 signum representation of the symmetric group 264 set of surjection sequences152 surface associated to a ribbon graph 292 category of surjections of finite sets60 category of unrooted trees with legs labeled by {0,...,n}250 category of labeled rooted n-trees52 category of X-labeled rooted trees52 tree corresponding to a surjection f151 operad for associative algebras with unit267 operad for commutative associative algebras with unit267316 operad for Poisson algebras with unit267 set of vertices of a graph 269 linear dual of a dg complex121 W-construction on an operad 28111 -homomorphism116 cyclohedron241 determinant operad128 dual of a cocycle 284 skew-symmetric Koszul sign196 composition (Gerstenhaber) product6 -complex125 desuspension of a graded vector space U314 Koszul sign195 -space101 category of stable S-labeled graphs of genus g271 free operad functor82 dual graph of a stable curve C304 free cyclic operad functor251 free K-operad functor88 universal bilinear form261 complex projective line207 triple for modular operads271 triple for -twisted modular operads281 cobar complex of an operad 20121125 dual dg operad of an operad 20121127 -order156 Koszul complex of an operad 121145 dual Koszul complex of an operad 146 categorial cobar complex of an operad 121149 reflection of a graded vector space U314 operad for associative algebras2044168 operad for -algebras20 bar construction on an operad 20 operad for Batalin — Vilkovisky algebras206257 operad for brace algebras31 operad for k-braid algebras27 operad for commutative associative monoids111 operad for commutative associative algebras2044168 operad for -algebras20 little intervals operad95 little squares operad95 little k-cubes operad1294203 little k-disks operad12203 Eilenberg — Zilber operad105 endomorphism operad643254 set of of isomorphism classes of n-trees with a special vertex234 free -algebra functor47 cofree nilpotent -coalgebra functor166 McClure — Smith operad3031 BRST complex24208 relative BRST complex26 operad for Leibniz algebras257 linear isometries operad106107 operad for Lie algebras2050 operad for -algebras18 operad for associative topological monoids2899 moduli space of n distinct labeled points in 212 moduli space of genus g Riemann surfaces with n distinct labeled points292305 moduli space of Riemann spheres with n+1 marked decorated points25 = presentation of an operad 139 -algebra46 -algebra chain complex173 -algebra cochain complex177 -algebra cohomology177 -algebra homology174 -coalgebra165 -degree177 -differential173 -homomorphism112 operad for Poisson algebras176256261 free -module functor139 semidirect product of an operad V with a group G204 quadratic dual of a quadratic operad 21 augmentation ideal of an augmented operad 187 pseudo-operad associated to an operad 110 operad 262 colored operad for homomorphisms of -spaces115 pseudo-operad of reduced rooted metric trees110 space of isomorphism classes of reduced rooted metric n-trees109221 non- pseudo-operad of reduced rooted planar metric trees112 space of isomorphism classes of reduced rooted metric planar n-trees112 pseudo-operad of reduced rooted trees55 set of isomorphism classes of reduced rooted n-trees55 non- operad of reduced planar rooted trees55109