An approach to the isotheory by means of extended pseudoisotopisms on 1 , Juan N´ nez 2 Ra´ ul Falc´ u˜ rafalgan@us.es 1 Department of Applied Mathematics I. University of Seville. 2 Department of Geometry and Topology. University of Seville. Rhodes. September 26, 2015. Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Outline 1 Preliminaries. 2 Extended isotopisms. 3 Extended pseudoisotopisms. Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Outline 1 Preliminaries. 2 Extended isotopisms. 3 Extended pseudoisotopisms. Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Preliminaries: Chronology 1942: Algebra isotopism (Albert). 1944: Quasigroup isotopism (Bruck). 1948: Lie-admissible algebras (Albert). 1967: Lie-Santilli admissibility (Santilli). 1978: Lie-Santilli isotheory (Santilli). 1983: Metric isospace (Santilli). 1992: Isoanalisis (Kadeisvili). 1993: Isotopology, isomanifold. (Tsagas-Sourlas). 1996: Isocalculus (Santilli). 2002: Isotopology (Tsagas-Sourlas-Santilli-Falc´ on-N´ u˜ nez). 2005: Isomanifolds based on generalized isotopisms (Falc´ on). Non-injective isoalgebras (Falc´ on, N´ u˜ nez). 2006: Santilli isotopisms of partial Latin squares (Falc´ on, N´ u˜ nez). 2007: Extended isotopisms of partial Latin squares (Falc´ on). 2015: Santilli autotopisms of partial groups (Falc´ on, N´ u˜ nez). Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Preliminaries: Algebra isotopisms (1942) Two algebras ( A 1 , · ) and ( A 2 , ◦ ) are isotopic if there exist three regular linear transformations α , β and γ from A 1 to A 2 such that α ( x ) ◦ β ( y ) = γ ( x · y ), for all x , y ∈ A 1 . Abraham Adrian Albert 1905-1972 The algebra A 2 is said to be isotopic to A 1 . The triple ( α, β, γ ) is said to be an isotopism between both algebras A 1 and A 2 . If α = β = γ , then this is an isomorphism . Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Preliminaries: Quasigroup isotopisms (1944) A quasigroup [Haussmann and Ore, 1937] is a nonempty set Q endowed with a product · , such that if any two of the three symbols u , v and w in the equation u · v = w are given as elements of Q , then the third is uniquely determined as an element of Q . Two quasigroups ( Q 1 , · ) and ( Q 2 , ◦ ) are isotopic if there exist three bijections α , β and γ from Q 1 to Q 2 such that α ( x ) ◦ β ( y ) = γ ( x · y ), for all x , y ∈ Q 1 . Richard Hubert Bruck 1914-1991 A quasigroup with identity element is a loop . A principal loop isotopism between ( Q 1 , · ) and ( Q 2 , ◦ ) is an isotopism of the form x ◦ y = ( x · u ) · ( v · y ) , where u , v ∈ Q 1 . ( Q 2 , ◦ ) is a loop with unit element I = u · v . If ( Q 1 , · ) is a group, then x ◦ y = x · T · y , where T = I − 1 ∈ Q 1 . Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Preliminaries: Lie-admissible algebra (1948) A Lie algebra is an anticommutative algebra A that satisfies the Jacobi identity J ( x , y , z ) = ( xy ) z + ( yz ) x + ( zx ) y = 0, ∀ x , y , z ∈ A . Marius Sophus Lie 1842-1899 Commutator product : [ x , y ] = xy − yx . Minus algebra : A → ( A − , [ ., . ]). Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Preliminaries: Lie-admissible algebra (1948) An algebra A over a field F is said to be Lie-admissible if the related minus algebra A − is a Lie algebra. Abraham Adrian Albert 1905-1972 Associative algebras. Lie algebras with product xy = x · y − y · x . Quasi-associative algebras ( λ -mutations ): ch ( F ) � = 2 and λ ∈ F ⇒ A ( λ ) with product ( x , y ) λ = λ · xy + (1 − λ ) · yx . [ x , y ] λ = (2 λ − 1) · [ x , y ] . A is Lie-admissible ⇔ A ( λ ) is Lie-admissible for all λ . Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Preliminaries: Santilli admissibility (1967) ( λ, µ ) -mutation of an algebra A : ( x , y ) λ,µ = λ · xy + µ · yx , where λ, µ ∈ F \ { 0 } and λ � = µ . Hence, [ x , y ] λ,µ = ( λ − µ ) · [ x , y ] Ruggero Maria Santilli Born in 1935 Lie-admissible algebras → Physics. Classical level for non-conservative systems: In pseudo-Hamiltonian Mechanics [Duffin, 1962] � � � n λ ∂ x ∂ y + µ ∂ x ∂ y [ x , y ] λ,µ = . ∂ q i ∂ p i ∂ p i ∂ q i i =1 Quantum-mechanical level for elementary-particle interacting or decaying regions: Fixed parameters: (Non) observability of quarks. Variable parameters : Indetermination principle. Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Preliminaries: Lie-Santilli isotheory (1978) Operator-deformations of an algebra A : ( x , y ) P , Q = xPy − yQx , where P , Q ∈ A . Ruggero Maria Santilli Born in 1935 P = Q ⇒ The algebra with product x × Q y = xQy is Lie-admissible. Classical mechanics (modified Hamilton’s equations): 2 n � ∂α i S ij ( t , α ) ∂ H ∂ x ∂α j = dx ( α ) ( x , H ) = , dt i , j =1 where α is a local chart in a manifold, H is a Hamiltonian and S ij is a non-singular C ∞ -tensor in a region. Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Preliminaries: Lie-Santilli isotheory (1978) Operator-deformations of an algebra A : ( x , y ) P , Q = xPy − yQx , where P , Q ∈ A . Ruggero Maria Santilli Born in 1935 P � = ± Q ⇒ The algebra with product ( x , y ) P , Q is also Lie-admissible. [ x , y ] P , Q = x ( P + Q ) y − y ( P + Q ) x . Hadronic mechanics (generalization of Heisenberg equation): ( x , H ) = xPH − HQx , where P and Q are non-Hermitian and non-singular operators in a physical system that represent non-self-adjoint forces. It makes possible to consider extended particles that admits additional contact, non-potential and non-Hamiltonian interactions. Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Preliminaries: Lie-Santilli isotheory (1978) Operator-deformations of an algebra A : ( x , y ) P , Q = xPy − yQx , where P , Q ∈ A . Ruggero Maria Santilli Born in 1935 P � = ± Q ⇒ The algebra with product ( x , y ) P , Q is also Lie-admissible. [ x , y ] P , Q = x ( P + Q ) y − y ( P + Q ) x . Question : How the product in an associative algebra can be modified to yield as general as possible a Lie-admissible algebra? Study of isotopisms that preserve axioms of Lie-admissibility. Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Preliminaries: Lie-Santilli isotheory (1978) Operator-deformations of an algebra A : ( x , y ) P , Q = xPy − yQx , where P , Q ∈ A . Ruggero Maria Santilli Born in 1935 P � = ± Q ⇒ The algebra with product ( x , y ) P , Q is also Lie-admissible. [ x , y ] P , Q = xTy − yTx . Question : How the product in an associative algebra can be modified to yield as general as possible a Lie-admissible algebra? Study of isotopisms that preserve axioms of Lie-admissibility. Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Preliminaries: Lie-Santilli isotheory (1978) A ≡ Associative algebra. T ≡ Isotopic element . Ruggero Maria Santilli Born in 1935 In the more general version, T can be taken outside the algebra A . In quantum mechanics, T depends on the state space determined by a set S of parameters: x , v , t , δ, . . . . T : S → A \ { 0 } . Isoproduct : � × : A × A × S → A ( x , y , s ) → x � × s y = xT ( s ) y . Lie-isotopic product : ˆ [ ., . ˆ ] : A × A × S → A ( x , y , s ) → ˆ [ x , y ˆ ] s = xT ( s ) y − yT ( s ) x = x � × s y − y � × s x . Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Preliminaries: Lie-Santilli isotheory (1978) A ≡ Associative algebra. S ≡ State space. T ( s ) ≡ Isotopic element . x � × s y = xT ( s ) y . ˆ [ x , y ˆ ] s = x � × s y − y � × s x . Ruggero Maria Santilli Born in 1935 Lie-Santilli isotheory : Step-by-step axiom-preserving construction of the conventional formulation of Lie theory in terms of the isoproduct � × s : Numbers, spaces, algebras, groups, symmetries, ... In Quantum mechanics, deformation of associative algebras are based on non-Hamiltonian effects, characteristics and interactions. Question : How to represent non-Hamiltonian terms in an invariant way? Ra´ ul Falc´ on, Juan N´ u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
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