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| Schurmann M. — White Noise on Bialgebras |
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| Предметный указатель |
Adapted process 57 58
Additive left white noise 85 88
Additive right white noise 85 88
Additive white noise 24
Algebra 12 15
Ampliation 46 81
Annihilation operator 109
Annihilation process 57 108
Anti-symmetric independence 17
Antipode 30
Araki — Woods embedding theorem 103
Azema martingale 81 93 137
Azema noise 94 137
Backward equation 84 85 89
Backward integral 68 84
Basic formula 52 53
Bialgebra 29 30
Bialgebra,  - 29
Bialgebra, cocommutative 103 112
Bialgebra, Graded 103
Bialgebra, involutive 30
Bialgebra, involutive - 30
Bialgebra, symmetric 30
Bose Fock space 41
Bose independence 17
Bose quantum Brownian motion 94
Bose quantum Wiener process 92 94
Brownian motion, quantum 81 94 137
Canonical commutation relations 110
Central limit theorem 130
Coalgebra 12
Coalgebra stochastic integral equation 65
Coalgebra, cocommutative 29 104
Coalgebra, Graded 15 29
Coassociativity 12
Coboundary 55 115
Cocommutative bialgebra 103 112
Cocommutative coalgebra 29 104
Cocycle 55 115
Coefficient algebra 31
Coefficient algebra of matrix semi-group 31
Coefficient algebra, deformation of 32
Coefficient algebra, non-commutative analogue of 32
Coherent state 41
Commutation factor 19
Commutation factor,  - 19
Commutation rules 17
Comodule algebra 14
Comodule coalgebra 15
Comodule vector space 13
Comultiplication 12
Conditionally positive linear functional 36
Conditionally positive sesquilinear form 76
Convolution product of linear functionals 27
Convolution product of linear mappings 27
Convolution semi-group 36
Counit 12
Creation operator 109
Creation process 57 108
Deformation of Heisenberg group 34 73
Deformation of Lie algebra of SL(2) 35
Deformation of SU(2) 34
Derivation component 117
Derivation component, maximal 117
Differential second quantization 41 57
Differential second quantization operator 110
Distribution of a quantum random variable 16
Distribution, 1-dimensional 16
Double module *-algebra 15
Double module algebra 15
Double module coalgebra 15
Double module vector space 14
Embedding theorem 112
Enveloping Hopf algebra 31
Equivalent processes 16
Even element 14 104
Even linear functional 21 76
Exponential element 30
Exponential vector 41
Factorization property 17
Fermi Fock space 111
Fermi independence 17
Fermi quantum Brownian motion 94
Fermi quantum Wiener process 94
Fermi Wiener process 94
Fock space, Bose 41
Fock space, Fermi 111
Forward equation 84 85 89
Forward integral 68
Fundamental theorem on coalgebras 26 96
Gelfand — Naimark — Segal (GNS) construction 16
Gelfand — Naimark — Segal (GNS) representation 103 107 112
Gelfand — Naimark — Segal (GNS) triplet 90 112
Generator of white noise 37
GNS see "Gelfand — Naimark — Segal"
Graded *-algebra 15
Graded *-vector space 15
Graded algebra 14
Graded coalgebra 15
Graded vector space 13
Graduation 21
Group 13
Group algebra 31 69
Group, involutive 15
Group-like element 30
Heisenberg group 32
Heisenberg group, deformation of 34 73
Hermitian linear functional 36
Hermitian operator 16
Hermitian sesquilinear form 76
Hochschildt cohomology 115
Homogeneous element 14
Hopf algebra 30
Hopf algebra, - 30
| Hopf algebra, enveloping 31
Hopf algebra, involutive 30
Hudson — Parthasarathy-integral 97 98
Independence 17
Independence, - 18
Independence, anti-symmetric 17
Independence, left 18 21
Independence, right 19 21
Independence, symmetric 17
Infinitely divisible representation 113
Infinitely divisible state 111
Infinitesimal generator 37
Integral sum identity 43
Interpolation 94
Invariance principle 132
Invariant 22
Involutive bialgebra 30
Involutive group 15
Involutive vector space 15
Ito's formula 57 62
Kernel 45
Kernel integral 59
Kernel valued-process 57 58
Left independence 18 21
Left invariant derivation 116
Left white noise 72
Lie *-algebra 31 113
Lie element 30
Loqually square integrable process 97
Maassen kernel 41
Matrix *-bialgebra over a finite group 34
Maximal derivation component 117
Maximal quadratic component 118
Module algebra 14
Module coalgebra 15
Module vector space 13
Multiplicative white noise 25 73 94
Non-commutative coefficient algebra 32
Non-commutative Poisson process 92 122
Odd operator 110
Opposite bialgebra 30
Parity operator 18 104
Poisson process 81 92 101 122
Positive linear functional 15
Positive sesquilinear form 75
Preservation operator 110
Preservation process 81 108
Primitive element 30
Property (C') 119
Property (C) 119
Property (D) 119
Pyramidal ordering 37 38
Quadratic component 118
Quadratic component, maximal 118
Quadratic linear functional 114
Quadratic linear functional, degenerate 115
Quantum Brownian motion 81 94 137
Quantum Ito's formula 57 62
Quantum Poisson process 81 92 122
Quantum probability space 15
Quantum random matrix 24
Quantum random variable 16
Quantum random vector 23
Quantum stochastic differential equation 65 68
Quantum stochastic integral 58 59 62
Quantum stochastic process 16
Quantum white noise 35
Quantum Wiener process 92 94
Reconstruction theorem 40
Representation theorem for while noise, general case 84
Representation theorem for while noise, symmetric case 56 68
Representative function 26 94
Right independence 19 21
Right white noise 72
Schoenberg correspondence 74
Second quantization 82
Second quantization process 81 82 103
Semi-direct product 69
Semi-group 30
Semi-group algebra 30
State 15
Symmetric bialgebra 30
Symmetric independence 17
Symmetric white noise 73
Symmetrization 69
Tensor algebra 23 31
Tensor algebra, graded 24
Tensor algebra, involutive 24
Tensor Hopf algebra 31
Tensor product, - 28
Tensor product, continuous 41
Tensor product, symmetric 41
Time-reversed white noise 38
Underlying additive white noise 88 103 130
Unitary group 25 32
Unitary q-white noise 25 101
Unitary white noise 25 99
VACUUM 54
Vector space 12
Vector space,  -graded 106
Vector space, graded 13
Vector space, involutive 15
Weyl algebra 123
White noise 35
White noise, - 35
White noise, additive 24
White noise, left 72
White noise, multiplicative 25
White noise, quantum 35
White noise, right 72
White noise, unitary 25 99 101
Wiener process 91 101
Wiener process, Bose 92
Wiener process, Fermi 94
Wiener process, quantum 92 94
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