Dissertation > Excellent graduate degree dissertation topics show

Researches on Operad and Associated Algebras

Author: ZhangYong
Tutor: LiFang;GuoZuo
School: Zhejiang University
Course: Basic mathematics
Keywords: operd Koszul totally compatible dialgebra tridendriform algebra bi-module free algebra Rota-Baxter algebra matching dialgebra matched pair Semi-direct sum pre-Lie algebra linearly compatible dialgebra Homotopy Transfer Theo-rem cobar construction self-dual composition rewriting method homotopy retract
CLC: O189.2
Type: PhD thesis
Year: 2013
Downloads: 3
Quote: 0
Read: Download Dissertation


The main results of this paper are divided into four parts.Firstly, we introduce the definition and equivalent definitions of an algebraic op-erad. In particular, we give the definitions of symmetric operad and nonsymmetric operad and show that they can be derived from each other. For the combinatorial defi-nition of the operad, we take the nonsymmetric operad for example. Moreover, we also introduce the relation between other types of operads and the trees in the combinatorial definition, such as cyclic operads and shuffle operads and so on. Next, we give the rela-tionship between the operad and homotopy theory and mainly show that how to obtain the homotopy algebras in operad theory. At last, we give some examples to introduce the new developments of the operad.Secondly, we discuss the concepts of a totally compatible dialgebra and a totally compatible Lie dialgebra, defined to be a vector space with two binary operations that satisfy individual and mixed associativity conditions and Lie algebra conditions respec-tively. We show that totally compatible dialgebras are closely related to bimodule alge-bras and semi-homomorphisms. More significantly, Rota-Baxter operators on totally compatible dialgebras provide a uniform framework to generalize known results that Rota-Baxter related operators give tridendriform algebras. We also show that a Rota-Baxter operator on a totally compatible Lie dialgebra gives rise to a PostLie algebra, generalizing the fact that a Rota-Baxter operator on a Lie algebra gives rise to a PostLie algebra. In our paper, both free totally compatible dialgebras and free Rota-Baxter to-tally compatible dialgebras are constructed guided by the general principle of universal algebra.Thirdly, we study the operad of linearly compatible di-algebras, denoted by As2, which is a nonsymmetric operad encoding the algebras with two binary operations that satisfy individual and sum associativity conditions. We also prove that the operad As2is exactly the Koszul dual operad of the operad2As encoding totally compatible di-algebras. We show that the operads As2and2As are Koszul by rewriting method. We make explicit the Homotopy Transfer Theorem for As2-algebras. Finaly, we give a systematic study of matching dialgebras corresponding to the op-erad As(2) in [91] as the only Koszul self dual operad there other than the operads of as-sociative algebras and Poisson algebras. The close relationship of matching dialgebras with semi-homomorphisms and matched pairs of associative algebras are established. By anti-symmetrizing, matching dialgerbas are also shown to give compatible Lie al-gebras, pre-Lie algebras and PostLie algebras. By the rewriting method, the operad of matching dialgebras is shown to be Koszul and the free objects are constructed in terms of tensor algebras. At last, we generalize the Koszul self-duality of the operad As(2) Moreover, the operadic complex computing the homology of the matching dialgebras is made explicit.

Related Dissertations

  1. Effects of Low Temperature Stress on Fatty Acid Composition and Content of Oreochromis Hornorum,S917.4
  2. The Research on Junior High School Students in Lack of Authenticity in Writing in Guangzhou Area,G633.34
  3. Distribution of the Contents of Pectin and Cellulose and Their Effect on Smoking Quality in Flue-cured Tobacco of China,S572
  4. Studies on Hay Making and Storage Technologies of Alfalfa and Perennial Ryegrass in Tibet,S816.53
  5. The Pharmacological Properties of Nicotinic Acetylcholine Receptor in Pardosa Pseudoannulata, an Important Predatory Enemy Spider of Insect Pests,S476.2
  6. Coparative Analyses of Carbon and Nitrogen Metabolism of Flue-cured Tobacco Growing in Henan and Yunnan,S572
  7. Study on the Effect of Creatine Monohydrates on Carcass Composition, Meat Quality and Energy Metabolism of Postmortem Muscle in Duck,S834.5
  8. Effect of Different Protein Sources on Growth Performance, Energy Budget and Hematology Index of Juvenile Epinephelus Fuscoguttatus,S963
  9. The Consideration and Practice Concerning Junior Literacy Education of Elementary School,G633.34
  10. The Problems and Strategies of Composition Teaching in the Senior High School in the Background of New Curriculum Reform,G633.34
  11. Study on Key Technology Optimization of Flue-cured Tobacco by Bulk Curing,TS44
  12. Effects of Climate Factors for Different Transplanting Dates on Quality and Ripe Tissue in Flue-cured Tobacco,S572
  13. Region Division and Quantitative Characteristics of Climate Factors for Qujing Tobacco-growing Areas,S572
  14. Effects of Different Gas Composition and Packaging Materials on Postharvest Quality and Active Oxygen Metabolism of Flammulina Velutipes,S646.15
  15. Study on the Physiological Responses of Three Marine Macroalgae to the Change of Nutrients Supplies in Seawater,S917.3
  16. Fundamental Studies on the Chemical Constituents of Brassica Rapa L.seed.,R284
  17. An Approach for Business Process Oriented Service Dynamic Composition,TP393.09
  18. Predicting Model of Coal Fired Fly Ash Resistivity,X701.2
  19. Studies on Chemical Constituents of the Aerial Parts of Triosteum Pinnatifidum,R284
  20. Effects of Modified Atmosphere Storage on the Quality of Walnut Kernels and Processed Products,TS255.6
  21. Community Ecology and Scientific Management of Lawn Weeds Communities in Taiyuan,S451

CLC: > Mathematical sciences and chemical > Mathematics > Geometry, topology > Topology ( the situation in geometry ) > Algebraic Topology
© 2012 www.DissertationTopic.Net  Mobile