1. Introduction ................................................. 1
2. Motivations .................................................. 9
2.1. Examples from Biology ................................... 9
2.1.1. Asexual propagation .............................. 9
2.1.2. Gametic algebras in asexual inheritance ......... 10
2.1.3. The Wright-Fisher model ......................... 11
2.2. Examples from Physics .................................. 12
2.2.1. Particles moving in a discrete space ............ 12
2.2.2. Flows in a discrete space (networks) ............ 12
2.2.3. Feynman graphs .................................. 13
2.3. Examples from Topology ................................. 15
2.3.1. Motions of particles in a 3-manifold ............ 15
2.3.2. Random walks on braids with negative
probabilities ................................... 15
2.4. Examples from Probability Theory ....................... 16
2.4.1. Stochastic processes ............................ 16
3. Evolution Algebras .......................................... 17
3.1. Definitions and Basic Properties ....................... 17
3.1.1. Departure point ................................. 17
3.1.2. Existence of unity elements ..................... 22
3.1.3. Basic definitions ............................... 23
3.1.4. Ideals of an evolution algebra .................. 24
3.1.5. Quotients of an evolution algebra ............... 25
3.1.6. Occurrence relations ............................ 26
3.1.7. Several interesting identities .................. 27
3.2. Evolution Operators and Multiplication Algebras ........ 28
3.2.1. Evolution operators ............................. 29
3.2.2. Changes of generator sets (Transformations of
natural bases) .................................. 30
3.2.3. "Rigidness" of generator sets of an evolution
algebra ......................................... 31
3.2.4. The automorphism group of an evolution
algebra ......................................... 32
3.2.5. The multiplication algebra of an evolution
algebra ......................................... 33
3.2.6. The derived Lie algebra of an evolution
algebra ......................................... 34
3.2.7. The centroid of an evolution algebra ............ 35
3.3. Nonassociative Banach Algebras ......................... 36
3.3.1. Definition of a norm over an evolution
algebra ......................................... 37
3.3.2. An evolution algebra as a Banach space .......... 38
3.4. Periodicity and Algebraic Persistency .................. 39
3.4.1. Periodicity of a generator in an evolution
algebra ......................................... 39
3.4.2. Algebraic persistency and algebraic
transiency ...................................... 42
3.5. Hierarchy of an Evolution Algebra ...................... 43
3.5.1. Periodicity of a simple evolution algebra ....... 44
3.5.2. Semidirect-sum decomposition of an evolution
algebra ......................................... 45
3.5.3. Hierarchy of an evolution algebra ............... 46
3.5.4. Reducibility of an evolution algebra ............ 49
4. Evolution Algebras and Markov Chains ........................ 53
4.1. A Markov Chain and Its Evolution Algebra ............... 53
4.1.1. Markov chains (discrete time) ................... 53
4.1.2. The evolution algebra determined by a
Markov chain .................................... 54
4.1.3. The Chapman-Kolmogorov equation ................. 56
4.1.4. Concepts related to evolution operators ......... 58
4.1.5. Basic algebraic properties of Markov chains ..... 58
4.2. Algebraic Persistency and Probabilistic Persistency .... 60
4.2.1. Destination operator of evolution algebra MX .... 60
4.2.2. On the loss of coefficients (probabilities) ..... 64
4.2.3. On the conservation of coefficients
(probabilities) ................................. 67
4.2.4. Certain interpretations ......................... 68
4.2.5. Algebraic periodicity and probabilistic
periodicity ..................................... 69
4.3. Spectrum Theory of Evolution Algebras .................. 69
4.3.1. Invariance of a probability flow ................ 69
4.3.2. Spectrum of a simple evolution algebra .......... 70
4.3.3. Spectrum of an evolution algebra at zeroth
level ........................................... 75
4.4. Hierarchies of General Markov Chains and Beyond ........ 76
4.4.1. Hierarchy of a general Markov chain ............. 76
4.4.2. Structure at the 0th level in a hierarchy ....... 77
4.4.3. 1st structure of a hierarchy .................... 80
4.4.4. kth structure of a hierarchy .................... 81
4.4.5. Regular evolution algebras ...................... 83
4.4.6. Reduced structure of evolution algebra MX ....... 86
4.4.7. Examples and applications ....................... 87
5. Evolution Algebras and Non-Mendelian Genetics ............... 91
5.1. History of General Genetic Algebras .................... 91
5.2. Non-Mendelian Genetics and Its Algebraic Formulation ... 93
5.2.1. Some terms in population genetics ............... 93
5.2.2. Mendelian vs. non-Mendelian genetics ............ 94
5.2.3. Algebraic formulation of non-Mendelian
genetics ........................................ 95
5.3. Algebras of Organelle Population Genetics .............. 96
5.3.1. Heteroplasmy and homoplasmy ..................... 96
5.3.2. Coexistence of triplasmy ........................ 98
5.4. Algebraic Structures of Asexual Progenies of
Phytophthora infestans ................................ 100
5.4.1. Basic biology of Phytophthora infestans ........ 101
5.4.2. Algebras of progenies of Phytophthora
infestans ...................................... 102
6. Further Results and Research Topics ........................ 109
6.1. Beginning of Evolution Algebras and Graph Theory ...... 109
6.2. Further Research Topics ............................... 113
6.2.1. Evolution algebras and graph theory ............ 113
6.2.2. Evolution algebras and group theory, knot
theory ......................................... 114
6.2.3. Evolution algebras and Ihara-Selberg zeta
function ....................................... 115
6.2.4. Continuous evolution algebras .................. 115
6.2.5. Algebraic statistical physics models and
applications ................................... 115
6.2.6. Evolution algebras and 3-manifolds ............. 116
6.2.7. Evolution algebras and phylogenetic trees,
coalescent theory .............................. 116
6.3. Background Literature ................................. 116
References .................................................... 119
Index ......................................................... 123
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