An approach to the isotheory by means of extended pseudoisotopisms

An approach to the isotheory by means of extended pseudoisotopisms R. M. Falcón and J. Núñez Citation: AIP Conference Proceedings 1738, 450002 (2016); doi: 10.1063/1.4952227 View online: http://dx.doi.org/10.1063/1.4952227 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1738?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Extended BRST Symmetries Quantum Approach AIP Conf. Proc. 1131, 17 (2009); 10.1063/1.3153443 Nuclear Matter Mean Field with Extended NJL Model AIP Conf. Proc. 660, 231 (2003); 10.1063/1.1570575 A perturbed‐mean‐field approach to the decay rates of excited vibrational states in extended systems: An application to I 2(Ne) n J. Chem. Phys. 100, 4355 (1994); 10.1063/1.466318 Analytical approach to molecular liquids. I. Site–site interaction model using an extended mean‐spherical approximation J. Chem. Phys. 91, 4861 (1989); 10.1063/1.456724 An extended mean spherical approximation for Coulombic systems J. Chem. Phys. 74, 3025 (1981); 10.1063/1.441426 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 150.214.182.15 On: Fri, 15 Jul 2016 10:19:01 An Approach to the Isotheory by Means of Extended Pseudoisotopisms R. M. Falcón∗and J. Núñez† ∗Dept of Applied Mathematics I, University of Seville (Spain) †Dept of Geometry and Topology. University of Seville (Spain) Abstract. Based on the traditional concept of isotopism, extended isotopisms were introduced by the authors in 2006 in order to provide a fundamental basis to the isotheory of Santilli. Since that first attempt, distinct studies on extended isotopisms have focused on the construction of partial Latin squares having a Santilli autotopism in their autotopism group. In order to deal with new structures, we introduce in this paper the concept of extended pseudoisotopism. This is based on the use of onto linear transformations that are not necessarily injective. Some examples are exposed throughout the paper. Keywords: Isotopism, isotheory PACS: 02.10.Ox, 11.10.Lm INTRODUCTION In 1942, Albert [1] introduced the concept of isotopism of algebras: Two algebras (A1,·)and (A2,◦)are isotopic if there exist three regular linear transformations α , β and γ from A1to A2such that α (u)◦ β (v) = γ (u·v),for all u,v∈A1.(1) The algebra A2is then said to be isotopic to A1and the triple Θ= ( α , β , γ )is said to be an isotopism between both algebras A1and A2. If α = β = γ , then this is an isomorphism. Since the original paper of Albert, a wide amount of papers have appeared in the literature that deal with isotopisms of distinct types of algebras as division [2, 3, 4, 5], Jordan [6, 7, 8], alternative [9, 10], absolute valued [11, 12], structural [13] and real two-dimensional commutative [14] algebras. Isotopism of Lie algebras were already considered by Albert himself [1] and, shortly after, in 1944, by Bruck [15]. More recently, in 1978, Santilli [16] took up the concept in the frame of a dynamical system based on an unitary Lie algebra Lendowed with an inner product ·, whose state space is determined by a set Sof parameters as coordinates, velocity, time, temperature or density, among others. He generalized the associative product u·vbetween Hermitian generators of the corresponding universal enveloping associative algebra by considering the new product uˆ·sv=u·ˆ T(s)·v,for all u,v∈Land s∈S,(2) where ˆ T:S→L\ {0}is called isotopic element1. The commutator product [u,v] = uˆ·sv−vˆ·supreserves the Lie axioms and is called the Lie-isotopic product. The application to Lie’s theory (enveloping algebras, Lie algebras and Lie groups) that emerges from this new product is the so-called Lie-Santilli isotheory [16, 17, 18, 19, 20, 21, 22]. For each state s∈S, there are also defined the elements ˆus=u·ˆ I(s),for all u∈L,(3) where ˆ I:S→L\{0}satisfies that i. ˆ T(s)·ˆ I(s)and ˆ I(s)·ˆ T(s)coincide with the identity element in L, for all s∈S. ii. The set ˆ Ls=L·ˆ I(s) = {ˆus:u∈L}coincides with L, for all s∈S. 1Note that, in order to clarify the relation that exists between the isotheory and the classical theory of isotopisms, the notation that is used throughout the current paper may differ from that used in [16] and its subsequent papers. International Conference of Numerical Analysis and Applied Mathematics 2015 (ICNAAM 2015) AIP Conf. Proc. 1738, 450002-1–450002-4; doi: 10.1063/1.4952227 Published by AIP Publishing. 978-0-7354-1392-4/$30.00 450002-1 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 150.214.182.15 On: Fri, 15 Jul 2016 10:19:01 iii. u·ˆ I(s)=v·ˆ I(s), for all s∈Sand u,v∈Lsuch that u=v. This map ˆ Iis called isounit, due to the fact that it generalizes the unit of the ground field of the original algebra. In particular, since the product ·is associative, the next equality holds ˆusˆ·sˆ I(s) = ˆus·ˆ T(s)·ˆ I(s) = ˆus=ˆ I(s)·ˆ T(s)·ˆus=ˆ I(s)ˆ·sˆus,for all u∈Land s∈S.(4) Further, ˆusˆ·sˆvs= (u·ˆ I(s))·ˆ T(s)·(v·ˆ I(s)) = (u·v)·ˆ I(s),for all u,v∈Land s∈S.(5) The set ˆ Ls=Lconstitutes, therefore, a Lie algebra with the commutator product applied to the product ˆ·s. It involves the definition of the isoproduct ˆ [ˆus,ˆvsˆ ]s=ˆusˆ·sˆvs−ˆvsˆ·sˆus= [u,v]ˆ·sˆ I(s),for all u,v∈L.(6) For each state s∈S, let α sbe the linear transformation from Lto itself, which is defined so that α s(u) = ˆus, for all u∈L. The identity (6) is then equivalent to ˆ [ α s(u), α s(v)ˆ ]s= α s([u,v]),for all u,v∈L.(7) The family of triples FS={( α s, α s, α s):s∈S}constitutes, therefore, a local state isomorphism of Lie algebras on the underlying dynamical system. Nevertheless, to the best knowledge of the authors, even if distinct papers and monographs have dealt with the foundations of the Lie-Santilli isotheory and its extension to other algebraic structures as groups, rings or vector spaces, among others [18, 19, 23, 24], there does not exist at this time any comprehensive study that reinterprets the Lie-Santilli isotheory by means of the theory of local state isomorphisms. A further study in this regard is, therefore, necessary. In 2006, in order to generalize the isotheory to a non-isomorphic frame and bring it closer to the classical theory of isotopisms, the authors used the concept of extended isotopism introduced in [25] as a way to reinterpret the dependence of the isounit on the state space of the underlying dynamical system as a family of classical isotopisms. At the time, distinct papers on extended isotopisms have focused on the construction of partial Latin squares having a Santilli isotopism in their autotopism group [26, 27, 28]. In order to deal with new algebraic structures, we generalize in this paper the concept of extended isotopism by reducing the condition of being injective. We introduce in this way the concepts of pseudoisotopism and extended pseudoisotopism, which constitute a new approach to lay the foundation for the isotheory. SANTILLI EXTENDED ISOTOPISMS The main strength of the isotheory consists of the dependence of the isounit on the state space of the underlying dynamical system. In the context of the Lie-Santilli isotheory, let us review how the authors reinterpreted this dependence in [26] in order to relate it with the classical theory of isotopisms. Let (L,·)be a Lie algebra, not necessarily unitary, associated to a dynamical system, whose state space is determined by a set Sof parameters. As a first step, we include a new parameter in the set Swith three possible states 1, 2 and 3. The new set of parameters is denoted as S=S×{1,2,3}. Let us consider a map ˆ I:S→L\{0}such that i. The set ˆ L(s,t)=L·ˆ I(s,t) = {u·ˆ I(s,t):u∈L}coincides with L, for all (s,t)∈S. ii. u·ˆ I(s,t)=v·ˆ I(s,t), for all (s,t)∈Sand u,v∈Lsuch that u=v. For each s∈S, let us consider the three linear transformations α s, β sand γ sfrom Lto itself so that α s(u) = u·ˆ I(s,1), β s(u) = u·ˆ I(s,2)and γ s(u) = u·ˆ I(s,3), for all u∈L. The conditions (i) and (ii) imposed to ˆ Iinvolve these three transformations to be onto and injective. As a consequence, the triple Θs= ( α s, β s, γ s)constitutes an isotopism between the Lie algebra (L,·)and the Lie algebra (L,ˆ·s), where ˆ·sis the product defined so that uˆ·sv= γ s( α −1 s(u)· β −1 s(v)),for all u,v∈L.(8) 450002-2 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 150.214.182.15 On: Fri, 15 Jul 2016 10:19:01 If α s= β s, then the triple Θsconstitutes indeed an isotopism between the Lie algebra Lendowed with the commutator product [u,v] = u·v−v·uand the Lie algebra Lendowed with the commutator product ˆ [u,vˆ ]s=uˆ·sv−vˆ·su= γ s( α −1 s(u)· α −1 s(v))− γ s( α −1 s(v)· α −1 s(u)) = γ s([ α −1 s(u), α −1 s(v)]),for all u,v∈L.(9) The family of triples FS={( α s, β s, γ s):s∈S}is called an extended Santilli isotopism of Lie-algebras. EXTENDED PSEUDOISOTOPISMS Extended isotopisms make possible to generalize the state isomorphism approach of the isotheory to a state isotopism approach. In the current section we generalize the latter by removing the injectivity in the conditions of the isounit ˆ I. A first attempt in this regard was already exposed by the authors in [26] for the construction of non-injective isoalgebras by means of Santilli isotopisms. We formalize the ideas exposed in that paper by introducing here the notions of pseudoisotopism and extended pseudoisotopism. The original idea on which both concepts are based was introduced in the Ph. D. Thesis of the first author [29]. Let (G1,·)and (G2,◦)be two groupoids, that is, a pair of sets G1and G2endowed with two respective binary operations ·:G1×G1→G1and ◦:G2×G2→G2. A triple Θ= ( α , β , γ )of onto maps from G1to G2is called a pseudoisotopism from (G1,·)to (G2,◦)if the following two conditions are satisfied i. γ (u·v) = γ (u′·v′),for all u,v,u′ ,v′∈G1such that α (u) = α ′(u)and β (v) = β ′(v). ii. α (u)◦ β (v) = γ (u·v),for all u,v∈G1. Observe that the second condition is consistent because of the first condition. If the three maps α , β and γ are injective, then the triple Θis a an isotopism. If α = β = γ , then Θis called a pseudoisomorphism. If there exist three elements u α ,u β and u γ in G1such that δ (u) = u·u δ , for all u∈G1and δ ∈ { α , β , γ }, then the triple Θis called a Santilli pseudoisotopism. Finally, if Iis a set of indices, then every family F={( α i, β i, γ i)}i∈Iformed by pseudoisotopisms from (G1,·)to (G2,◦)is called an extended pseudoisotopism. It is a Santilli extended pseudoisotopism if each triple of the family Fis a Santilli pseudoisotopism. Let us finish the paper with some examples on this point. Example 1 In the field of complex numbers C, let G be a group generated by i ∈C, endowed with the usual product ·in C, that is, G ={1,−1,i,−i}. Let us consider the subgroup H ={1,−1}of G and let α :G→H be such that α (1) = α (−1) = 1and α (i) = α (−i) = −1. The triple ( α , α , α )is a pseudoisomorphism from (G,·)to (H,·)that preserves the structure of group. ▹ Example 2 Let (R,+,×)be the field of real numbers and let U be the set of differentiable real functions of one real variable. This set constitutes an R-vector space with the natural operators (f+g)(x) = f(x)+g(x),for all f ,g∈U and x ∈R.(10) ( λ ·f)(x) = λ ×f(x),for all f ∈U and x ∈R.(11) Let us consider the map α :U→Rso that α (f) = ∂ f ∂ x(2), for all f ∈U. The latter is onto because, given a ∈R, the function f (x) = a 4x2satisfies that α (f) = a. However, α is not injective. To see it, it is enough to consider the functions f1(x) = 15x2and f2(x) = 5x3, for which α (f1) = α (f2) = 60. The triple Θ1= ( α , α , α )is a pseudoisomorphism from (U,+) to (R,+) because a+b=∪ f,g∈U{ α (f+g): ∂ f ∂ x(2) = a, ∂ g ∂ x(2) = b}. The triple Θ2= (Id, α , α )is a pseudoisomorphism from (U,·)to (R,×)where a×b=∪ f∈U{ α (a·f): ∂ f ∂ x(2) = b}. The pair (Θ1,Θ2)constitutes therefore a pseudoisotopism from (U,+,·)to (R,+,×).▹ 450002-3 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 150.214.182.15 On: Fri, 15 Jul 2016 10:19:01 Example 3 Let us consider the polynomial ring F2[x]over the finite field F2. Let us define the maps α 0and α 1from F2[x]to itself, so that α t(p) = p+t, for all polynomial p ∈F2[x]and t ∈ {0,1}. The family F= {( α t1, α t2, α (t1+t2) (mod 2))}t1,t2∈{0,1}is a Santilli extended isotopism from (F2[x],+) to itself. Specifically, p+q= α t1(p−t1)+ α t2(q−t2) = α (t1+t2) (mod 2)(p+q−((t1+t2) (mod 2))),for all t1,t2∈ {0,1}. ▹ CONCLUSIONS AND FURTHER STUDIES Extended (pseudo)isotopisms are introduced here to extend the state isomorphism frame in which isotheory is comprehended to a more general state (pseudo)isotopism frame. A further study on both concepts is necessary. Specifically, a study that focuses on Santilli (pseudo)isotopisms of partial Latin rectangles can be an interesting starting point in this regard. ACKNOWLEDGMENTS Authors are very grateful to Prof. Santilli for his valuable help in our research on the isotheory and congratulate him for his 80 birthday. REFERENCES 1. A. A. Albert, Ann. of Math. (2) 43, 685–707 (1942). 2. R. Sandler, Amer. J. Math. 84, 239–264 (1962), ISSN 0002-9327. 3. H. P. Petersson, Abh. Math. Sem. Univ. Hamburg 35, 215–222 (1971), ISSN 0025-5858. 4. G. Benkart, and J. M. Osborn, “Real division algebras and other algebras motivated by physics,” in Proceedings of the Third Workshop on Lie-Admissible Formulations (Univ. Massachusetts, Boston, Mass., 1980), Part A, 1980/81, vol. 4, pp. 392–443, ISSN 0162-5519. 5. E. Dieterich, J. Algebra Appl. 4, 517–538 (2005), ISSN 0219-4988. 6. R. H. Oehmke, and R. Sandler, J. Reine Angew. Math. 216, 67–87 (1964), ISSN 0075-4102. 7. K. McCrimmon, Trans. Amer. Math. Soc. 159, 445–468 (1971), ISSN 0002-9947. 8. K. McCrimmon, Scripta Math. 29, 229–236 (1973), collection of articles dedicated to the memory of Abraham Adrian Albert. 9. K. McCrimmon, Math. Ann. 191, 253–262 (1971), ISSN 0025-5831. 10. M. Babikov, Proc. Amer. Math. Soc. 125, 1571–1575 (1997), ISSN 0002-9939. 11. A. A. Albert, Ann. of Math. (2) 48, 495–501 (1947), ISSN 0003-486X. 12. J. A. Cuenca Mira, E. Darpö, and E. Dieterich, Bull. Sci. Math. 134, 247–277 (2010), ISSN 0007-4497. 13. B. N. Allison, and W. Hein, J. Algebra 69, 120–142 (1981), ISSN 0021-8693. 14. Z. Balanov, and Y. Krasnov, Comm. Algebra 31, 4571–4609 (2003), ISSN 0092-7872. 15. R. H. Bruck, Trans. Amer. Math. Soc. 56, 141–199 (1944), ISSN 0002-9947. 16. R. M. Santilli, Hadronic J. 1, 223–423. (1978). 17. R. M. Santilli, Nuovo Cimento 51, 570–576 (1967). 18. J. V. Kadeisvili, Acta Appl. Math 50, 131–165 (1998). 19. R. M. Falcón, and J. Núñez, Fundamentos de la isoteoría de Lie-Santilli, International Academic Press. America-Europe-Asia, 2001. 20. R. M. Santilli, Nuovo Cimento B 121, 443–486 (2006). 21. P. Nikolaidou, and T. Vougiouklis, Ratio Matematica 26, 113–128 (2014). 22. R. M. Falcón, J. Núñez, and A. Aversa, Algebra, Groups and Geometries 32, 135–308 (2015). 23. G. T. Tsagas, and D. S. Sourlas, Mathematical Foundations of the Lie-Santilli Theory, Hadronic Press, 1993. 24. J. V. Kadeisvili, Rend. Circ. Mat. Palermo serie II 42, 83–136 (1996). 25. R. M. Falcón, Electron. Notes Discrete Math. 29, 503–507 (2007). 26. R. M. Falcón, and J. Núñez, Hadronic J. 29, 285–298 (2006). 27. R. M. Falcón, and J. Núñez, J. Dyn. Syst. Geom. Theor. 5, 19–32 (2007). 28. R. M. Falcón, and J. Núñez, Am. J. Mod. Phys. In press (2015). 29. R. M. Falcón, Isovariedades isodiferenciables y grupos de Lie-Santilli, Ph.D. thesis, University of Seville (2005). 450002-4 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 150.214.182.15 On: Fri, 15 Jul 2016 10:19:01