Evolution Algebras and their Applications

1. Introduction 2. Motivations 2.1. Examples from Biology 2.1.1 Asexual propagation 2.1.2. Gametic algebras in asexual inheritance 2.1.3. The Wright-Fisher model 2.2. Examples from Physics 2.2.1. Particles moving in a discrete space 2.2.2. Flows in a discrete space (networks) 2.2.3. Feynman graphs 2.3. Examples from Topology 2.3.1. Motions of particles in a 3-manifold 2.3.2. Random walks on braids with negative probabilities 2.4. Examples from Probability Theory 2.4.1. Stochastic processes 3. Evolution Algebras 3.1. Definitions and Basic Properties 3.1.1. Departure point 3.1.2. Existence of unity elements 3.1.3. Basic definitions 3.1.4. Ideals of an evolution algebra 3.1.5. Quotients of an evolution algebra 3.1.6. Occurrence relations 3.1.7. Several interesting identities 3.2. Evolution Operators and Multiplication Algebras 3.2.1. Evolution operators 3.2.2. Change of generator sets (Transformations of natural bases) 3.2.3. 'Rigidness' of generator sets of an evolution algebra 3.2.4. The automorphism group of an evolution algebra 3.2.5. The multiplication algebra of an evolution algebra 3.2.6. The derived Lie algebra of an evolution algebra 3.2.7. The centroid of an evolution algebra 3.3. Non-associative Banach Algebras 3.3.1. Definition of a norm over an evolution algebra 3.3.2. An evolution algebra as a Banach space 3.4. Periodicity and Algebraic Persistency 3.4.1. Periodicity of a generator in an evolution algebra 3.4.2. Algebraic persistency and algebraic transiency 3.5. Hierarchy of an Evolution Algebra 3.5.1. Periodicity of a simple evolution algebra 3.5.2. Semi-direct-sum decomposition of an evolution algebra 3.5.3. Hierarchy of an evolution algebra 3.5.4. Reducibility of an evolution algebra 4. Evolution Algebras and Markov Chains 4.1. Markov Chain and Its Evolution Algebra 4.1.1. Markov chains (discrete time) 4.1.2. The evolution algebra determined by a Markov chain 4.1.3. The Chapman-Kolmogorov equation 4.1.4. Concepts related to evolution operators 4.1.5. Basic algebraic properties of Markov chains 4.2. Algebraic Persistency and Probabilistic Persistency 4.2.1. Destination operator of evolution algebra M(X) 4.2.2. On the loss of coefficients (probabilities) 4.2.3. On the conservation of coefficients (probabilities) 4.2.4. Certain interpretations 4.2.5. Algebraic periodicity and probabilistic periodicity 4.3. Spectrum Theory of Evolution Algebras 4.3.1. Invariance of a probability flow 4.3.2. Spectrum of a simple evolution algebra 4.3.3. Spectrum of an evolution algebra at zero-th level 4.4. Hierarchies of General Markov Chains and Beyond 4.4.1. Hierarchy of a general Markov chain 4.4.2. Structure at the 0-th level in a hierarchy 4.4.3. 1-th structure of a hierarchy 4.4.4. k-th structure of a hierarchy 4.4.5. Regular evolution algebras 4.4.6. Reduced structure of evolution algebra M(X) 4.4.7. Examples and applications 5. Evolution Algebras and Non-Mendelian Genetics 5.1. History of General Genetic Algebras 5.2. Non-Mendelian Genetics and Its Algebraic Formulation 5.2.1. Some terms in population genetics 5.2.2. Mendelian versus non-Mendelian genetics 5.2.3. Algebraic formulation of non-Mendelian genetics 5.3. Algebras of Organelle Population Genetics 5.3.1. Heteroplasmy and homoplasmy 5.3.2. Coexistence of triplasmy 5.4. Algebraic Structures of Asexual Progenies of Phytophthora infestans 5.4.1. Basic bi