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THE NEW ISO-, GENO-, AND HYPER-MATHEMATICS OF HADRONIC MECHANICS

Original content uploaded February 15, 1997. Revisions uploaded on February 22 and March 29, April 4, and June 15, 1997 thanks to numerous critical comments by various visitors, which are acknowledged with gratitude. Additional critical comments should be sent to ibr@gte.net and will be appreciated.

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This section lists open research problems in pure and applied mathematics. All interested mathematicians in all countries, including graduate students, are welcome to participate in the research.

Following the introductory section, individual open problems are presented via:

a) a brief summary of the topic.
b) a statement of the open problem(s) suggested for study.
c) the motivation for the proposed research.
d) the suggested IBR member(s) and/or Editor(s) for technical assistance.
e) representative references.

I. INTRODUCTION Studies initiated in the late 1970's under the support of the Department of Energy at Department of Mathematics of Harvard University by the theoretical physicists Ruggero Maria Santilli have indicated that current mathematical knowledge is generally dependent on the assumption of the simplest conceivable unit e which assumes either the numerical value e = +1, or the n-dimensional unit form e = diag. (1, 1, . . . , 1).

Systematic studies were then initiated for the reformulation of contemporary mathematical structures with respect to a generalized unit E of the same dimension of the original unit e (e.g., an nxn matrix) but with an arbitrary functional dependence on a local chart r, its derivatives with respect to an independent variable (e.g., time t) v = dr/dt, a = dv/dt, and and any other needed variable,

(1) e -> E = E(t, e, v, a, . . . ),

Jointly, the conventional associative product axb among generic quantities a, b (e.g., numbers, vector fields, operators, etc.) is lifted into the form

(2) axb -> a*b = axTxb, T = 1/E,

(3) e -> E(t, r, v, a, ... ), ab -> a*b = axTxb, E = 1/T,

By conception and construction, the new formulations are locally isomorphic to the original ones for all positive-definite generalized units E > 0. As a result, maps (3) do not yield "new mathematical axioms", but only "new realizations" of existing mathematical axioms and, for this reason, they were called "isotopic" in the Greek meaning of being "axiom-preserving".

When E is no longer Hermitean (e.g., it is nowhere singular and real-valued but non-symmetric), then we have the general loss of the original axioms in favor of more general axioms (see below for examples) and, for this reason maps (3) were called "genotopic" from the Greek meaning of being "axiom inducing". In this case we have two different units <E = 1/R and E> = 1/S, generally interconnected by the conjugation <E = (E>)Ý with corresponding ordered products to the left and to the right,

(4a) <E = 1/R, a<b = AxRxb,

(4b) E> = 1/S, a>b = AxSxb, R = SÝ.

Additional classes of mathematical structures occur when the generalized units are multivalued, or subjected to anti-isomorphic conjugation (see below).

The new lines of mathematical inquiries emerged from these studies imply novel formulations of: number theory, functional analysis, differential geometries, Lie’s theory, topology, etc. For example, ordinary numbers and angles, conventional and special functions and transforms, differential calculus, metric spaces, enveloping algebras, Lie algebras, Lie groups, representation theory, etc., must be all reformulated under isotopies for the generalized product a*b = axTxb in such a way to admit E(t, r,v, a, ... ) = 1/T as the new left and right unit, and a more general setting occurs under genotopies.

To illustrate the nontriviality of these .liftings it is sufficient here to recall that Lie's theory with familiar product [A, B] = AxB -- BxA (where A, B are vector fields on a cotangent bundle or Hermitean operators on a Hilbert space, and AxB, BxA are conventional associative products), is linear, local-differential and potential-Hamiltonian, thus possessing clear limitations in its applications.

The isotopies and genotopies of Lie's theory , called Lie-Santilli isotheory and genotheory, respectively, include the corresponding liftings of universal enveloping algebras, Lie algebras, Lie groups, transformation and representation theories, etc. and are based on the following corresponding generalized products first proposed by Santilli in 1978

(5a) [A, B]* = A*B - B*A = AxT(t, r, ...)xB - BxT[t, r,...)xA, T = TÝ,

(5b) (A, B) = A<B - B>A = AxR(t, r, ...)xB - BxS(t, r, ...)xA, R = SÝ,

As expected, the theories with products (5a) and (5b) have been proved to provide an effective characterization of nonlinear, nonlocal and nonhamiltonian systems of increasing complexity (the former applying for stable-reversible condiions, and the latter for open-irreversible conditions, see the next Web Page 19). Their consistent treatment requires corresponding new mathematics, called iso- and genlo-mathematics, respectively. For instance, it would be evidently inconsistent to define an algebra with generalized unit E(t, r, ...) = EÝ over a conventional field of numbers with trivial unit e = +1, and the same happens for functional analysis, differential calculus, geometries, etc.

The studies initiated by Santilli were continued by numerous scholars including Gr. Tsagas, D. S. Sourlas, J. V. Kadeisvili, H. C. Myung, S. Okubo, S. I. Vacaru, B. Lin, D. Rapoport-Campodonico, R. Ohemke, A. K. Aringazin, M. Nishioka, G. M. Benkart, A. Kirukhin, J. Lohmus, J. M. Osborn, E. Paal, L. Sorgsepp, N. Kamiya, P. Nowosad, D. Juriev, C. Morosi, L. Pizzocchero,R.Aslander, S. Keles,and other scholars. A comprehensive list of contributions in related fields up to 1984 can be find in Tomber's Bibliography in Nonassociative Algebras, C. Baltzer et al., Editors, Hadronic Press, 1984. A bibliography on more recent contributions can be found in the monograph by J. Lohmus, A. Paal and L. Sorgsepp, Nonassociative Algebras in Physics, 1994, Hadronic Press (see Advanced Titles in Mathematics in this Web Site).

The mathematical nontriviality of the above studies is also illustrated by the fact that, at a deeper analysis, isotopies and genotopies imply the existence of SEVEN DIFFERENT LIFTINGS of current mathematical structures with a unit, each of which possess significant subclasses, as per the following outline:

1) ISODUAL MATHEMATICS. It is characterized by the so-called isodual map, first introduced by Santilli in 1985 (see [I-1] for a recent account), given the lifting of a generic quantity a (a number, vector-field, operator, etc.) into its anti-Hermitean form

(6) a -> isod(a) = - aÝ,

(7) e = 1 -> E = - 1, axb -> a*b = (-bÝ)x(-1)x(-aÝ) = - bÝxaÝ.

Since the norm of isodual numbers is negative-definite, isodual mathematics has resulted to be useful for a novel representation of antimatter (see Page 19).

The visitor should be aware that contemporary mathematics appears to be inapplicable for a physically consistent representation of antimatter at the CLASSICAL level, with corresponding predictable shortcomings at the particle level. In fact,we only have today one type of quantization, e.g., the symplectic quantization. As a result, the operator image of contemporary mathematical treatments of antimatter does not yield the needed charge conjugate state. At any rate, the map from matter to antimatter must be anti-automorphic (or, more generally,anti-isomorphic), as it is the case for charge conjugation in second quantization.

The only known map verifying these conditions at all levels of treatment is Santilli's isodual map (6). This yields a novel classical representation of antimatter with a corresponding novel isodual quantization which does indeed yield the correct charge conjugate state of particles (see Web Page 19). Thus, the isodual mathematics resolves the historical lack of equivalence in the treatment between matter and antimatter according to which the former is treated at all levels, from classical mechanics to quantum, field theories, while the latter was treated only at the level of second quantization.

To understand the implications, the visitor should keep in mind that contemporary mathematics does not appear to be applicable for an effective treatment of antimatter, thus requiring its reconstruction in an anti-isomorphic form.

2, 3) ISOTOPIC MATHEMATICS AND ITS ISODUAL. The isotopic mathematics or isomathematics for short [I-1] is today referred to formulations for which the generalized unit E, called isounit, has a nontrivial functional dependence and it is Hermitian, E = E(t, r, v, ...) = EÝ. An important case is the Lie-Santilli isotheory with basic isoproduct (5a).

The isodual isotopic mathematics is the image of the isomathematics under maps (6) and therefore has the unit isod(E) = -EÝ = -E.

These structures have been classified by the theoretical physicist J. V. Kadeisvili in 1991 into:

CLASS I, when E is positive-definite (isotopies),
CLASS II when E is negative definite (isodualities),
CLASS III, given by the union of Classes I and II,
CLASS IV, including the preceding classes plus null values of E (or singular values of T), and
CLASS V, when E is arbitrary, e.g., a step-function or a distribution.

4, 5) GENOTOPIC MATHEMATICS AND ITS ISODUAL. The isotopies were proposed in 1978 as particular cases of the broader genotopies [I-1], which are characterized by two different generalized units <E = 1/R and E> = 1/S for the genomultiplication to the left and to the right according to Eqs. (4). The resulting genotopic mathematics, or genomathematics for short, is given by a duplication of the isomathematics, one for ordered products to the left and the other to the right.

The isodual genomathematics is the isodual image of the preceding one, and it is characterized by the systematic application of map (6) to each of the left and right genomathematics.

In an evening seminar delivered at ICM94 Santilli proved that the genotopies can also be axiom-preserving and can therefore provide a still broader realization of known axioms. The proof was presented for product (5b) which, when considered on ordinary spaces and fields with the conventional unit e, is known to verify Albert’s axiom of Lie-admissibility. The same product was proved to verify the Lie axiom when each of the two terms A<B and B>A is computed in the appropriate genoenvelope and genofield with the corresponding genounit.

6, 7) HYPERSTRUCTURAL MATHEMATICS AND ITS ISODUAL. At the IBR meeting on multivalued hyperstructures held at the Castle Prince Pignatelli in August 1995, the mathematician Thomas Vougiouklis and R. M. Santilli presented a new class of hyperstructures, those with well defined hyperunits characterized by hyperoperations. A subclass of the latter hyperstructures important for applications is that with hyperunits characterized by ordered sets of non-Hermitean elements,

(8a) <E = {<A, <B, ...} = 1/R = {1/R1, 1/R2, ...},

(8b) E> = {A>, B>, ... } = 1/S = {1/S1, 1/S2, ...},

Needless to say, the above studies are in their first infancy and so much remains to be done.

The material of this Web Page is organized following the guidelines of memoir [I-1] according to which there cannot be really new applications without really new mathematics, and there cannot be really new mathematics without new numbers. We shall therefore give utmost priority to the lifting of numbers according to the above indicated seven different classes. All remaining generalized formulations can be constructed from the novel base fields via mere compatibility arguments.

We shall then study the novel spaces and geometries which can be constructed over the new fields because geometries have the remarkable capability of reducing the ultimate meaning of both mathematical and physical structures to primitive, abstract, geometric axioms.

We shall then study: the generalized functional analysis which can be constructed on the preceding structures, beginning from Kadeisvili’s new notions of continuity; the all fundamental Lie-Santilli theory; the underlying novel Tsagas-Sourlas integro-differential topology; and other aspects.

Only primary references with large bibliography are provided per each section. Subsequent calls to references of preceding sections are indicated with [I-1], [II-1], etc.

REFERENCE OF SECT. 1: We recommend to study the following memoir and some of the large literature quoted therein

[I-1] R. M. Santilli, Nonlocal-integral isotopies of differential calculus, mechanics, and geometries, Rendiconti Circolo Matematico Palermo, Supplemento No. 42, pages 7-83, 1996.

See also

[I-2] J. V. Kadeisvili, An Introduction to the Lie-Santilli Isotopic theory, Mathematical Methods in Applied Sciences, Vol. 19, pages 1349-1395, 1996.

[I-3] J. V. Kadeisvili, "Santilli's Isotopies of Contemporary Algebras, Geometries and Relativities, Second Edition, Ukraine Academy of Sciences, Kiev, 1997 (First Edition 1992).

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II: OPEN RESEARCH PROBLEMS IN NUMBER THEORY

Prepared by the IBR staff.

DEFINITION II.1: Let F = F(a,+,x) be a conventional field of (real R, complex C or quaternionic Q) numbers a with additive unit 0, multiplicative unit e = 1, sum a+b and product axb. Santilli’s isodual field [II-1] isodF = isodF(isoda, isod+,isodx) is a ring of elements isoda = -aÝ, called isodual numbers, with isodual sum (isoda)isod+(isodb) = (-aÝ-bÝ) and isodual multiplication (isoda)isodx(isodb) = (-aÝ)(-1)(-bÝ) = -(aÝ)(bÝ) under which the additive unit is isisod0 = 0 and the multiplicative unit is isode = -e = -1.

LEMMA II-1 [II-1]: The isodual field is a field (i.e., it verifies all axioms of a field).

PROPOSITION II-1 [loc. cit.]: The map F -> isodF is anti-isomorphic.

PROPOSED RESEARCH II-1: Study the isodual number theory, including theorems on prime, factorization, etc.

SIGNIFICANCE: Isodual numbers have a negative norm, thus being useful to represent antimatter.

FOR TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

DEFINITION II-2: Let F = F(a,+,x) be a conventional field of (real R, complex C or quaternionic Q) numbers a with additive unit 0, multiplicative unit e = 1, sum a+b and product axb. Santilli’s isofields [II-1] F* = F*(a*,+*,x*) are rings of elements a* =axE called isonumbers, where a is an element of F, axE is the multiplication in F, and E = 1/T is a well behaved, everywhere invertible, Hermitean and positive-definite quantity (e.g., a matrix, vector field, operator, etc. of Kadeisvili’s Class I) generally outside the original field F, equipped with the isotopic sum a*+*b* = (a+b)xE and product a*x*b* = a*xTxb* = (axb)xE, with additive unit 0* = 0 and multiplicative unit E, called isounit. Santilli’s isodual isofields isodF*(isoda*,isod+*,isodx*) are the anti-Hermitean images of F* under the isodualities of the elements of F* and all its operations.

LEMMA II-2 [II-2]: Isofields verify all axioms of a field (including closure under the combined associative and distributive laws). The lifting F -> F* is therefore an isotopy.

PROPOSITION II-2 (ref.[II-1], p. 284): When E is an element of the original field F (e.g., an ordinary real number for F = R), the isofield F*(a,+*,x*) is also a field (i.e., closure occurs for conventional numbers a without need to use the isonumbers a* = axE).

PROPOSED RESEARCH II-2: Formulate the real isonumber theory: 1) with a basic unit given by an arbitrary, positive, real number E = n > 0; and 2) under isoduality to a negative-definite unit isodE = -n < 0. These problems can be studied via the simplest possible class of Santilli isofields R*(a,+*,x*) and their isoduals isodR*(isoda,isod+*,isodx*) in which the elements a are not lifted, as per Proposition II.2 above. The study implies the re-inspection of all conventional properties of number theory in order to ascertain which one is dependent on the selected unit. As an example, it is known that the notion of prime depends on the selected unit [II-1] because, e.g., the number 4 becomes prime for the isounit E = 3.

SIGNIFICANCE: An important advance of memoir [II-1] is that the axioms of a field need not to be restricted to the simplest possible unit +1 dating back to biblical times, because they equally hold for arbitrary units. This basic property has far reaching implications. In mathematics the property implies the lifting of all structures defined on numbers; in physics the broadening of the unit implies basically novel applications (See the subsequent Web Page 19 on Open Research Problems in Physics); and in biology it implies a structural revision of current theories (see the subsequent Web Page 20 on Open Research Problems in Biology).

FOR TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

DEFINITION II-3: Let F = F(a,+,x) be a conventional field of (real, complex C or quaternionic Q) numbers a with additive unit 0, multiplicative unit e = 1, sum a+b and product axb. Santilli’s genofields to the right [II-1] F> = F>(a>,+>,x>) are rings of elements a> = axE> called genonumbers, where a is an element of F, axE> is the multiplication in F, and E> = 1/S is a well behaved, everywhere invertible and non-Hermitean quantity generally outside F, equipped with all operations ordered to the right, i.e., the ordered genosum to the right (a>)+>(b>) = (a+b)xE>, the ordered genoproduct to the right (a>)>(b>) = (a>)xSx(b>) = (axb)xE>, etc., conventional additive unit to the right 0> = 0 and generalized (left and right) unit E> for the multiplication to the right called genounit. Santilli’s genofields to the left <F(<a,<+,<) are rings with genonumbers <a = <Exa, all operations ordered to the left, such as genosum (<a)<+(<b) = <Ex(a+b), genoproduct (<a)<(<b) = <Ex(axb), etc., with genoadditive unit to the left <0 = 0 and multiplicative genounit to the left <E = 1/R which is generally different than that the right. The isodual left and right genofields are the isodual images of the left and right genofields.

REMARKS: In the definition of fields and isofields there is no ordering of the multiplication in the sense that in the products axb and a*b one can either select a multiplying b from the left , a>b or b multiplying a from the right a<b, because a>b = a<b (even for non-commutative fields such as quaternions). A genofield requires that all multiplications and related operations (e.g., division, etc.) be ordered EITHER to the right OR to the left because now, e.g., for a commutative field F = R or C, we have the properties a>b = b>a and a<b = b<a, but in general a>b ‚ a<b. Note that in each case the genounit is the left and right unit because (E>)>(a>) = (a>)>(E>) = a> for all possible a>.

LEMMA II-3 [II-1]: Each individual genofield to the right F> or to the left <F is a field. Thus each lifting F -> F> and F -> <F is an isotopy.

PROPOSITION II-3 [II-1]: When E> and <E are elements of an ordinary field F, each genofield F>(a,+>,x>) and <F(a,<+,<x) is a field.

PROPOSED RESEARCH II-4/II-5: Formulate the number theory with 1) a basic unit given by a positive real number E> = n in which all operations are ordered to the right; 2) formulate the same theory under an ordering to the left with a different positive-definite genounit <E = b; 3) construct the isoduals of both theories. These problems can be studied via the simplest possible class of Santilli genofields F>(a,+>,x>) and <F(a,<+,<x) in which the elements are not lifted, as per Proposition II-3 above. The study implies the re-inspection of all conventional properties of the isonumber theory.

SIGNIFICANCE: Another significant advance of memoir [II-1] is that the axioms of a field, not only do not need the restriction to the unit +1, but the operations can be all restricted to be EITHER to the right OR to the left. This simple property has additional far reaching mathematical, physical and biological implications. In mathematics, it implies a dual lifting of all isotopic structures. In physics it implies an axiomatic representation of the irreversibility of the physical world via the most fundamental mathematical notion, the unit. In fact, operations ordered to the right can represent motion forward in time, while operations ordered to the left can represent motion backward in time.

Irreversibility is then reduced to the differences between E> and <E or, equivalently, between a>b and a<b. The inclusion of the isodualities implies the capability to represent all possible four different motions in time: motion forward to future time E>, motion backward to past time <E, motion forward from past time isod(<E), and motion backward from future time isod(E>). In theoretical biology, Santilli’s genonumbers are the foundation of the first known consistent mathematical representation of the irreversibility of biological structures. The addition of isoduality then permits the mathematical representation of certain bifurcations in biology whose treatment is simply beyond conventional mathematics.

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

Repeat the studies of Problem II-4/II-5 but for the case when the genounits to the right and to the left and given by an ordered set of nonhermitean invertible elements equipped, first, with conventional and, then, with hyperoperations. For a definition of hyperfields with conventional operations see ref. [I-1] of Sect. I. For their broader definition with hyperoperations see ref. [II-2] below.

SUGGESTED TECHNICAL ASSISTANCE: Consult

Prof. T. Vougiouklis
Department of Mathematics
Democritus University of Thrace
GR-67100 Xanthi, Greece, fax +30-551-39348, or

Prof. M. Stefanescu
Department of Mathematics
Ovidius University, Bd. Mamaia 124
Costanta 8700, Romania,

REFERENCES OF SECT. II:

[II-1] R. M. Santilli, Isonumbers and genonumbers of dimension 1, 2, 4, 8, their isoduals and pseudoduals, and “hidden numbers” of dimension 3,5,6,7, Algebras, Groups and Geometries Vol. 10, pages 273-322, 1993 [II-2] T. Vougiouklis, Editor, New Frontiers in Hyperstructures, Hadronic Press, 1996.

RECENT PAPERS APPEARED IN THE RESEARCH PROPOSED IN SECT. II

ON SANTILLI'S ISOTOPIES OF THE THEORY OF REAL NUMBERS,
COMPLEX NUMBERS, QUATERNIONS AND OCTONIONS
N. Kamiya
Department of Mathematics
Shimane University
Matsue 690, Japan
"New Frontiers in Algebras, Groups and Geometries",
Gr. Tsagas, Editor, Hadronic Press, Palm Harbor (1996), pp. 523-551.

A CHARACTERIZATION OF PSEUDOISOFIELDS
N. Kamiya
Department of Mathematics
Shimane University
Matsue 690, Japan
and
R. M. Santilli
Institute for Basic Research
P., O. Box 1577
Palm harbor, FL 34682, U.S.A.
ibr@gte.net
"New Frontiers in Algebras, Groups and Geometries",
Gr. Tsagas, Editor, Hadronic Press, Palm Harbor (1996), pp. 559-570.

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III: OPEN PROBLEM IN GEOMETRIES

Prepared by the IBR staff.

DEFINITION III-1: Let S = S(r,g,R) be a conventional n-dimensional metric or pseudo-metric space with local chart r, nowhere singular, real-valued and symmetric metric g and invariant (rt)xgxr (where rt denotes transposed of r) over a conventional field R = R(a,+,x) of real numbers. Santilli’s isodual spaces isodS = isodS(isodr,isodg,isodR) is the vector space with local chart isodr = -r, isodual metric isodg = -g and isodual invariant (isodrt)x(isodg)x(isodr) = [(rt)xgxr](isode) on isodR. Isaodual geometries are the geometries on isodual spaces, thus based on negative-definite units.

PROPOSITIONS III-1 [I-1]: Isodual spaces are anti-isomorphic to the original space.

PROPOSED RESEARCH III-1: Study the isodual Euclidean, isodual Minkowskian, isodual Riemannian, isodual symplectic and other isodual geometries, including the isodual calculus, the isodual sphere (i.e., the sphere with negative radius), the isodual light cone, etc. [I-1].

SIGNIFICANCE: Isodual geometries are fundamental for the recent isodual representation of antimatter, e.g., to characterize the shape of an antiparticle with negative units.

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

PROBLEMS III-2/III-3: STUDIES ON ISOGEOMETRIES AND THEIR ISODUALS

DEFINITION III-2: Let S = S(r,g,R) be a conventional n-dimensional metric or pseudo-metric space with local chart r, nowhere singular, real-valued and symmetric metric g and invariant (rt)xgxr (where rt denotes transposed) over a conventional real field R = R(a,+,x). Santilli’s isospaces [I-1] S*(r*,G*,R*) are vector spaces with local isocoordinates r* = rxE, isometric G* = g*xE = TxgxE and isoinvariant (r*t)*G*r* = [(rt)x(g*)xr]xE over an isofield R* = R*(a*,+*,x*) with a common isounit E = 1/T of Kadeisvili Class I. Santilli’s isogeometries [III-1] are the geometries of isospaces. The isodual isospaces and isodual isogeometries are the corresponding images under isoduality [I-1,III-1].

LEMMA III-2 [I-1,III-1]: Isospaces S*(x*,G*,F*) are locally isomorphic to the original space S(x,g,F). The lifting S -> S* is therefore an isotopy.

Proof. Each component of the metric g is lifted by the corresponding element of T, while the unit is lifted by the corresponding inverse amount E = 1/T, thus preserving the original geometric axioms.q.e.d.

PROPOSITION III-3 [III-2]: The axioms of the Euclidean geometry in n-dimension admit as particular cases all possible well behaved, real-valued, symmetric and positive-definite metrics of the same dimension.

REMARKS.To be consistently defined, Santilli’s isogeometries require the isotopies of the totality of the mathematical aspects of the original geometry, all formulated for a common isounit E with the same dimension of the isospace. This requires, not only the isotopies of fields and vector spaces, but also those of all other aspects.

Provide a mathematical formulation of Santilli’s isoeuclidean, isominkowskian, isoriemannian, isosymplectic and other isogeometries and their isoduals which have been only preliminarily studied for physicists in ref. [III-1].

SIGNIFICANCE: The mathematical and physical implications are significant indeed. Mathematically, the studies permit advances such as: the unification of all geometries of the same dimension into one single isotope; the admission under the Riemannian axioms of metric with arbitrary, nonlinear, integro-differential dependence in the velocities and other variables; the representation of nonhamiltonian vector-fields in the local chart of the observer (see the alternative to Darboux’s theorem, ref. [I-1], p. 63 motivated by the fact that, in view of their nonlinearity, Darboux’s transforms cannot be used in physics because the transformed frames cannot be realized in experiments and, in any case, they violate the axioms of Galilei’s and Einstein’s special relativity due to their highly noninertial character). Physically, the studies permit truly basic advances, such as the first quantitative research on the origin of the gravitational field, a geometric unification of the special and general relativity via the isominkowskian geometry in which the isometric is a conventional Riemannian metric, a novel operator formulation of gravity verifying conventional quantum axioms; and other advances (see Web Pages 19 and 20).

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net, or

Prof. D. Sourlas
Department of Mathematics
University of Patras
Gr-26100 patras, Greece
Fax +30-61-991 980

Prof. Gr. Tsagas
Department of Mathematics
Aristotle University
Thessaloniki 54006, Greece
Fax +30-31-996 155

Prof. R. Miron
Department of Mathematics
“Al. I. Cuza” University
6600 Iasi, Romania
rmiron@uaic.ro

DEFINITION III-3: Let S = S(r,g,R) be a conventional n-dimensional metric or pseudo-metric space with local chart r, nowhere singular, real-valued and symmetric metric g and invariant (rt)xgxr over a conventional real field R = R(a,+,x). Santilli’s n-dimensional genospaces to the right [I-1] S>(r>,G>,R>) are vector spaces with local genocoordinates to the right r> = rxE>, genometric G> = (g>)x(E>) = SxgxE>, genoinvariant (r>t)>(G>)(r>) = [(rt)x(g>)xr]xE> over the genofield R> = R>(a>,+>,x>), common genounit to the right E> = 1/S given by an everywhere invertible, real-valued, non-symmetric nxn matrix, and all operations ordered to the right. Santilli’s genogeometries [III-1] are the geometries of genospaces. The isodual genospaces and isodual genogeometries are the corresponding images under isoduality [I-1,III-2]. Santilli’s n-dimensional genospaces to the left [I-1] <S(<r,<+,<R) are genospaces over genofields with all operations ordered to the left and a common nxn-dimensional genounit to the left <E = 1/R which is generally different than that to the right E> = 1/S. Genogeometries to the left and their isoduals are the geometries over the corresponding genospaces.

LEMMA III-3: Genospaces to the right S> and, independently, those to the left <S, are locally isomorphic to the original spaces S. Proof. The original metric g is lifted in the form g -> Sxg, but the unit is lifted by the inverse amount I -> E = 1/S, thus preserving the original axioms. q.e.d.

PROPOSED RESEARCH: Provide a mathematical formulation of Santilli’s genoeuclidean, genominkowskian, genoriemannian, genosymplectic and other genogeometries to the left, their corresponding forms to the right and their isoduals which have been preliminarily studied in ref. [III-1] for physicists.

SIGNIFICANCE: Another important aspect of memoir [I-1] is that the Riemannian axioms do not necessarily need a symmetric metric because the metrics can also be nonsymmetric with structure g> = Sxg, S nonsymmetric, provided that the geometry is formulated on an isofield with isounit given by the INVERSE of the nonsymmetric part, E = 1/S, and the same occurs for the left case. This property has permitted the first quantitative studies on irreversibility of interior gravitational problems via the conventional Riemannian axioms, as it occurs in the physical reality, e.g., the irreversibility of the structure of Jupiter or of a collapsing star, for which purpose the genogeometries were constructed in the first place [III-1].

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Extend the studies of the genogeometries to the right and to the left to the case when the corresponding genounits are given by an ordered set of invertible, real-valued and nonsymmetric nxn elements, first, with ordinary operations and then with hyperoperations.

The existence of these geometries has been only indicated in ref. [I-1] without any detailed treatment.

SUGGESTED TECHNICAL ASSISTANCE: Consult the IBR staff at ibr@gte.net

REFERENCES OF SECT. III:

[III-1] R. M. Santilli, Elements of Hadronic Mechanics, Vol. I, Mathematical Foundations, Ukraine Academy of Sciences, Kiev, Second Edition, 1995.

RECENT PAPERS APPEARED IN THE RESEARCH PROPOSED OF SECT. III

ISOAFFINE CONNECTION AND SANTILLI'S ISORIEMANNIAN METRIC ON AN ISOMANIFOLD
Gr. Tsagas
Department of Mathematics
Aristotle University
Thessaloniki 54006, Greece
Algebras, Groups and Geometries, Vol. 13, pages 149-169, 1996

STUDIES ON SANTILLI'S LOCALLY ANISOTROPIC AND INHOMOGENEOUS
ISOGEOMETRIES, I: ISOBUNDLES AND GENERALIZED ISOFINSLER GRAVITY
Sergiu I. Vacaru
Institute of Applied Physics
Academy of Sciencves of Moldova
5,. Academy Street
CHISINAU 2028, REPOUBLIC OF MOPLDOVA
Fax +3732-738149, E-address lises@cc.acad.md
In press at Algebras Groupos and Geometries, Vol. 14, 1997

CARTAN'S STRUCTURE EQUATIONS ON SANTILLI-TASGAS-SOURLAS ISOMANIFOLDS
Recept Aslander
Inonu Universitesi
Egitim Fakultesi
Matematik Egitimi Bolumu
44100 Malatya, Turkey
and
Sadik Keles
Inonu Universitesi
Fen-Edebiyat Fakultesi
Matematik Bolumi
44100 Malatya, Turkey
In press at Algebras, Groups and Geometries, Vol. 14, 1997

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IV: OPEN RESEARCH PROBLEMS IN FUNCTIONAL ANALYSIS

Conventional and special functions and transforms and functional analysis at large are dependent on the assumed basic unit. As an example, a change of the two-dimensional unit of the Gauss plane implies a change in the very definition of angles and trigonometric functions, and the same happens for hyperbolic functions, Fourier, Laplace and other transforms, Dirac and other distributions, etc.

To be operational, the seven classes of novel mathematical methods of the preceding sections require seven corresponding generalized forms of functional analysis, which are here recommended for study in a progressive way, beginning with the simplest possible case of isoduality.

Mathematical work done to date in these new topics has been rather limited. We here mention: Santilli has preliminarily studied the structure of the seven forms of differential calculus [I-1], isotrigonometric and isohyperbolic functions, the isofourier transforms and few other aspects {III-1]; Kadeisvili [IV-1,2] has studied basic definitions of isocontinuity and its isodual and reinspected some of the studies in the field; H. C. Myung and R. M. Santilli [IV-3] studied the isotopies of the Hilbert space, Dirac delta distributions and few other notions; A. K. Aringazin, D. A. Kirukhin and R. M. Santilli [IV-4] have studied the isotopies of Legendre, Jacobi and Bessel functions and their isoduals; M. Nishioka [IV-5] studied the Dirac-Myung-Santilli isodelta distribution (see [III-1] for a review up to 1995).

SUGGESTED TECHNICAL ASSISTANCE: Contact Prof. J. V. Kadeisvili at ibr@gte.net, or A. K. Aringazin and D. A. Kirukhin at aringazin@kargu.krg.kz

REFERENCE FOR SECT. IV:

[IV-1] J. V. Kadeisvili, Elements of functional isoanalysis, Algebras, Groups and Geometries vol. 9, pages 283-318, 1992.
[IV-2] J. V. Kadeisvili, Elements of the Fourier-Santilli isotransforms, Algebras, Groups and Geometries Vol. 9, pages 319-242, 1992
[IV-3] H. C. Myung and R. M. Santilli, Modular-isotopic Hilbert space formulation of the exterior strong problem, Hadronic J. Vol. 5, pages 1277-1366, 1982.
[IV-4] A. K. Aringazin, D. A. Kirukhin and R. M. Santilli, Isotopic generalization of Legendre, Jacobi and Bessel functions, Algebras, Groups and Geometries Vol. 12, pages 255-359, 1995.
[IV-5] M. Nishioka, Extension of the Dirac-Myung-Santilli delta functions to field theory, Lett. Nuovo Cimento Vol. 39, pages 369-372, 1984.

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V. STUDIES ON THE LIE-SANTILLI ISO-, GENO- AND HYPER-THEORIES AND THEIR ISODUALS

Prepared by the IBR staff

DEFINITION V-1: Let L be an n-dimensional Lie algebra with ordered Hermitean basis X = {A,B, ...} = XÝ, conventional commutator [A, B] = AxB - BxA (where AxB is conventionally associative) over a field F (of characteristics zero). A Lie-Santilli isodual algebra [I-1] isodL is the image of L under the isodual map (5), thus including isodual generators isodX = -XÝ = -X, isodual commutator isod[A, B] = (isodA)isodx(isodB) - (isodB)isodx(isodA) = - [A, B], etc., all defined on an iusodualF with negative-definite unit E = -Diag. (1, 1, ..., 1), and norm.

LEMMA V-1 [I-1]: IsodL is anti-isomorphic to L.

PROPOSED RESEARCH V-1: Reformulate Lie's theory (enveloping associative algebras, Lie algebras, Lie groups, transformation and representation theories, etc.) for the Lie-Santilli isodual theory with an n-dimensional negative-definite unit E = - Diag(1, 1, ... 1).

SIGNIFICANCE: Isodual symmetries characterize antiparticles.

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DEFINITION V-2: Let U(L) be the enveloping associative algebra of a Lie algebra L with infinite-dimensional basis (Poincare’-Birkhoff-Witt theorem) 1, X, (Xi)x(Xj )(i ¾ j), .. and conventional exponentiation expA = 1 + A/1! + AxA/2! + ...over a field F. Santilli’s universal enveloping isoassociative algebra U*(L) of L [V-1] of Class I is characterized by the isotopies of the Poincare’-Birkhoff-Witt theorem) with infinite-dimensional isobasis

(9) E, X, (Xi)*(Xj) (i ¾ j), ...,

where the isounit E = 1/T is of Kaidesvilki Class I (positive-definite) has the same dimension of the representation at hand and A*B = AxTxB, with base isofield F* characterized by the same isounit E of U*(L). The isodual isoassociative envelope is the image of U* under isoduality.

REMARKS. The conventional exponentiation is no longer applicable for U*(L) and must be replaced by the isoexponentiation

(10) isoexpA = E + A/1! + A*A/2! + ... = {exp(AxT)}xE = Ex{exp(TxA)}.

The symbol U*(L) rather than U*(L*) is used to indicate that the basis X of L is unchanged under isotopy and merely redefined in isospace,thus X* = X. This is the most significant case on physical grounds because generators of Lie symmetries represent quantities such as energy, linear momentum, angular momentum, etc. which, as such, cannot be changed. Only the operations defined in them can be changed.

Since Lie’s theory leaves invariant its basic unit, the functional dependence of the isounit E is left unrestricted and, therefore, can depend on an independent variable t (say time), local chart r of the carrier space, its derivatives v = dr/dt, a = dv/dt and any other needed variable, E = E(t, r, v, a, ...) = 1/T. The nontriviality of the Lie-santilli isotheory can therefore be seen up-front because it implies the appearance of a nonlinear, integro-differential element T in the EXPONENT of the group structure, Eq. (10).

DEFINITION V-3. Let L be an n-dimensional Lie algebra as per Definition V-1. A Lie-Santilli isoalgebra [V-1] L* of Class I is the algebra homeomorphic to the antisymmetric algebra [U*(L)]- of U*(L). It can be defined as an isovector space with the same ordered basic X* = X of L equipped with the isoproduct

(11) [A, B]* = A*B - B*A = AxT(t, r, v, a, ...)xB - BxT(t, r, v, a, ...)xA

REMARKS. Lie-Santilli isoalgebras verify the conventional Lie axioms (anti-commutativity of the product and Jacobi identify), only formulated in isospace (that is, with respect to the isoassociative produce A*B) over the isofield F*.

LEMMA V-3 [III-1]: Lie-Santilli isoalgebras are left and right isolinear, i.e., they verify the left linearity conditions on L* as an isovector space over F*,

(12a) {[(a*)*A* + (b*)*B*, C*]* = (a*)*[A*,B*]* + (b*)*[B*, C*]*

(12b) [(A*)*B*, C*] = (A*)*[B*, C*]* + [A*, C*]*(B*)

and corresponding right conditions. Isoalgebras L* are isolocal (in the sense of being everywhere local-differential except at the isounit E) and isocanonical (in the sense of admitting a canonical structure in isospace, see [III-1] for brevity).

DEFINITION V-4: Let G be an n-dimensional connected Lie transformation group r’ = K(w)xr on a space S(r,F), where w are the parameters in F, verifying the usual conditions (differentiability of the map GxS -> S, invariance of the basic unit e = I, and linearity), as well as the conditions to be derived from the Lie algebra L via exponentiation

(13a) Q(w) = {exp(ixXxw)} x Q(0) x {exp(-ixwxX)}

(13b) i [Q(dw) - Q(0) ] / dw = QxX - XxQ = [ Q, X].

A connected Lie-Santilli isotransformation group [V-1] G* of Class I is the set of isotransforms

(14) r*’ = Q*(w*)] * (r*) = [Q*[w*)] x T(t, r, v, a, ...) x r

on a Class I isospace S*(r*,F*), where now the isoparameters w* = wxE belong to F*, which verifies the usual conditions in their isotopic form (Kadeisvili’s isodifferentiability of the isomap (G*)*S* -> S*, invariance of the isounit E, and isolinearity), as well as the conditions to be derivable from the Lie-Santilli isoalgebra L* [V-1]

(15a) Q*(w*) = {isoexp[i(X*)*(w*)]} * [Q*(0)] * {isoexp[-i(w*)*(X*)]} =

= { exp (i X x T x w) } x Q*(0) x { exp(-iwTX) },

(15b) i [Q*(dw*) - Q*(0) ] / dw* = (Q*) *(X*) - (X*) * (Q*) = [ Q*, X*]*

with isogroup laws [V-1]

(16) [Q*(w*)]*[Q*(w*’)] = Q*(w* + w*’] , [Q*(w*)]*[Q*(-w*)] = Q*(0*) = E

The isodual Lie-Santilli isogroups isodG* are the isodual image of G* under map (5).

LEMMA V-3 [V-1]: Lie-Santilli isoenvelopes U*, isoalgebras L* and isogroup G* are locally isomorphic to the original structures U, L, and G, respectively for all possible positive-definite isounit E (not so otehrwise). The liftings

(17) U -> U*, L -> L* and G -> G*

are therefore isotopies.

PROPOSED RESEARCH V-2: Conduct mathematical studies on the Lie-Santilli isotheory of Class I and its isodual with particular reference to: the isostructure theory; the isorepresentation theory; and related aspects.

REMARKS. At the abstract, realization-free level, isoenvelopes U*, isoalgebras L* and isogroups G* coincide with the conventional envelopes U, algebras L and groups G, respectively, by conception and construction for all positive-definite isounits E (not necessarily so otherwise). This illustrates the insistence by Santilli in indicating that the isotopies do not produce new mathematical structures, but only new realizations of existing abstract axioms.

As a result of the, the isorepresentation theory of U, L and G on isospaces over isofields is expected to coincide with the conventional representations of the original structures U, L and G on conventional spaces over conventional fields. The aspect of the isorepresentation theory which is important for applications is the PROJECTION of the isorepresentation on conventional spaces. Stated differently, Lie’s theory admits only one formulation, the conventional one. On the contrary, the covering Lie-Santilli isotheory admits two formulations, one in isospace over isofield and one given by its projection on conventional spaces over conventional fields.

The latter are important for applications, e.g., because the physical space-time is the conventional Minkowski space, while the isominkowski space is a mathematical construction. As a result, the isorepresentation theory of the Poincare’-Santilli isosymmetry [V-4] on isominkowski space over isofields is expected to coincide with that of the conventional symmetry on the conventional Minkowski space over the conventional field of real numbers. The mathematically and physically significant aspects are given by the PROJECTION of the isorepresentation on the conventional Minkowski space-time.

SIGNIFICANCE: The isotheory characterizes all infinitely possible, well behaved, arbitrarily nonlinear, nonlocal-integral and nonhamiltonian, classical and operator systems by reducing them to identical isolinear, isolocal and isocanonical forms in isospaces over isofields, thus permitting a significant simplification of notoriously complex structures.

PROPOSED RESEARCH V-3: Study the Lie-Santilli isotheories of Classes III (union of Class I with positive-definite and II with negative definite isounits E), Class IV (Class III plus null isounit E) and Class V arbitrary isounits E, including discontinuous realzioations). As a particular case unify all simple Lie algebras of the same dimension in Cartan's classification into one single isoalgebra of the same dimension of Class III, whose study has been initiated by Tsagas and Sourlas [V-4].

REMARK 1. In his original proposal on the isotopies of Lie's theory of 1978 (see the references inn [V-1]), Santilli proved the loss at the abstract level of all distinction between compact and noncompact Lie algebras of the same dimension provided that the isounit has an arbitrary positive- or negative-definite signature (Class III). This was illustrated via the algebra of the rotation group in three dimension O(3). When its conventional generators X1, X2, X3 (the components of the angular momentum) are equipped with the isounit E = Diag. (+1, +1., -1) and isoproduct (11) they characterize the noncompact O(2.1) algebra. The isoalgebra O*(3) with the fixed generators X1, X2, X3 equipped with isoproduct (11) and a isotopic element T of Class III therefore unifies all simple Lie algebras of dimension 3. This result has been proved to hold also for all orthogonal and unitary algebras, and it is expected to hold for all possible Lie algebras, including the exceptional ones.

REMARK 2. As indicated in the subsequent Web Page 19, the zeros of the isounit represent gravitational singularities. The study of the Lie-Santilli theory of Class IV is therefore important for applications. No study in on record at this writing in this field which requires the prior study of numbers, spaces, geometries, etc., whose units can be psoitive, negative as wel as null. No study is also on record on the isotoppies of Class V.

IMPORTANT NOTE. Visitors of this page should be aware that the treatment of the isoproduct [A, B]* = AxTxB - BxTxA on conventional spaces over conventional fields is not invariant under the group action and, as such, it has no known physical value. In fact, when realized on a Hilbert space over a conventional field, isogroups G* are characterized by nonunitary transforms WWÝ ‚ I. As a result, the base unit of a conventional treatment of the isoproduct [A, B]* is not left invariant by the isogroup, I -> I’ = WxIxWÝ ‚ I, and, consequently, the isoproduct itself is not invariant, [A, B]* -> Wx[A, B]*xWÝ = A’xT’xB’ - B’xT’xA’, where T’ = (WÝ to -1)xTx(W to -1) ‚ T. The loss of the traditional invariance of Lie’s theory then implies the lack of meaningful applications.

On the contrary, when treated via the isotopic mathematics, that is, formulated on isospaces over isofields, the isoproduct [A, B]* is fully invariant. For instance, by considering again the operator realization, the originally nonunitary structure of G* is turned into identical isounitary forms, i.e., we can write W = (W*)x(square root of T) for which WxWÝ = (W*)*(W*Ý) = (W*Ý)*(W*) = E, in which vase the base isounit E of the isofield is invariant, E -> E’ = (W*)*E*(W*Ý) = (W*)*(W*Ý) = E, and the isoproduct is consequently invariant, (W*)*[A, B]*x(W*Ý) = A’xTxB’ - B’xTxA’, where one should note that T is numerically preserved. The above occurrence illustrate the necessity of using Santilli’s isonumbers and isospaces for meaningful applications.

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Prof. D. Sourlas
Department of Mathematics
University of Patras
Gr-26100 patras, Greece
Fax +30-61-991980

Prof. Gr. Tsagas
Department of Mathematics
Aristotle University
Thessaloniki 54006, Greece
Fax +30-31-996 155

HISTORICAL NOTE: Lie algebras are "nonassociative" in the sense that their product [A, B] = AxB - BxA is nonassociative. Yet their representations are reduced to those of their universal enveloping associative algebras U(L) with product AxB because of the homeomorphism between L and the attached antisymmetric algebra U(L)-.

In 1948 the American mathematician A. A. Albert (Trans. Amer. Mat. Soc. Vol. 64, p. 552, 1948) introduced the notion of Lie-admissible algebras (presented in Sect. I and called the First Condition of Lie-admissibility). The formulation was done within the context of nonassociative algebras, in which context they have been studied by various mathematicians until recently. Also, Albert was primarily interested in the "Jordan content" of a given "nonassociative" algebra and, for this reason, he studied the product (a, b) = pxaxb + (1-p)xbxa, where p is a real parameter, which admits the commutative Jordan product for p = 1/2.

In 1967 Santilli (Nuovo Cimento Vol 51, p. 570, 1967) was the first physicist to study Lie-admissible algebras. He noted that Albert's definition did not admit Lie algebras in their classification and, for this reason, the algebras had limitations in physical applications. He therefore introduced a new notion of Lie-admissibility which is Albert's definition plus the condition of admitting Lie algebras in their classification (this is called the Second Condition of Lie-admissibility). In particular, Santilli studied the product (a,b) = pxaxb - qxbxa, where p, q, and p+or-q are non-null parameters, which does indeed admit the conventional Lie product as a (nondegenerate) particular case, and which constitutes the first formulation in scientific record of the "deformations" of Lie algebras of the contemporary physical literature (see the next Web Page 19).

In 1978 Santilli (Hadronic J. Vol. 1, p. 574, 1978) notes his Second Condition of Lie-admissibility was still insufficient for physical applications because Lie-admissibility implies "nonunitary" time evolutions under which the "parameters" p and q become "operators". He therefore introduced a more general definition of Lie-admissibility which is Albert's definition plus the conditions that the attached antisymmetric algebra is Lie-isotopic, rather than Lie, and the algebras admit conventional Lie algebras in their classification (this is the Third Condition of Lie-admissibility, also called Albert-Santilli Lie-admissibility, or General Lie-admissibility).

In this way, Santilli introced the product (A, B) = AxRxB - BxSxA, Eq.s (5b), where R, S, and R+or-S are fixed and nonull vector-fields, matrices, operators, etc. for which the attached antisymmetric algebra is the isotopic form [A,B] = (A, B) - (B, A) = AxTxB - BxTxA, T = R+S, while admitting of the conventional Lie product for R+S = 1. The product (A, B) is also the first on scientific records of the so-called "quantum groups" of the contemporary physical literature. The same product, being neither totally antisymetric nor totally symmetric, includes as particular cases supersymmetric and other generalizations of the Lie product (see the next Web Page 19).

In the same memoir of 1978, Santilli reduced the study of the Lie-admissible product (A, B) = AxRxB - BxSxA to its two isoassociative envelopes AxRxB and BxSxA, that is, he reduced the representation theory of the nonassociative product (A, B) to that of its two, right and left envelopes with "isoassociative" product AxRxB and BxSxA [V-5], in essentially the same way as the study of the Lie product [A, B] = AxB - BxA is reduced to that of the associative ones AxB and BxA.

The terms Santilli's Lie-admissible theory or genotheory are referred to the latter context, that is, to a dual left and right lifting of Lie's theory (enveloping associative lagebras, Lie algebras, Lie groups, representation theory, etc.).

The tool which permitted this formulation is that of a bi-representation (split-null extension) [V.5]. The main point is that bi-modular Lie-admissible structures are contained in the structure of CONVENTIONAL Lie’s groups. In fact, Eq.s (13) can be written [V-1]

(18a) Q(w) = {exp(ixXxw)} > Q(0) < {exp(-ixwxX)}

(18b) i [Q(dw) - Q(0) ] / dw = W < X - X > W

where > means conventional modular-associative “action to the right” and < “action to the left”. The bi-modular character is trivial in Lie’s case because the action of a conventional Lie group from the left is minus the transpose action from the right.

An important observation of Ref. [V-1] is that group structure (18) can also be written in a non-trivial bi-modular form characterized, first, by the isotopic modular actions to the right and to the left and, then, their differentiation into genotopic forms. To put it bluntly, a bimodular Lie-admissible structure is already contained in the conventional structure of Lie groups. It merely remained un-noticed until 1978. In fact, the modular associative product to the right can be realized via the right genoassociative algebra U> with product A>B = AxSxB and that to the left via the left genoassociative algebra <U with product A<B = AxRxB with corresponding genounits to the right and left E> = 1/S and <E = 1/R. Eq.s (18) then yield Santilli’s Lie admissible theory [V-1]

(19) Q(w) = {exp>(ixXxw)} > Q(0) < {exp<(-ixwxX)} =

= {[exp(ixXxSxw)]xE>} x S x Q(0) x R x { (19 i [Q(dw) - Q(0) ] / dw = Q < X - X > Q =

= Q x R x X - X x S x Q = (Q, X)

In this way the representation theory of the Lie-admissible algebra with nonassociative product (A, B) is first reduced to a bi-representation theory on {<U, U>} and then shown to admit a Lie-admissible group structure in a way fully parallel to the conventional Lie case.

In an evening seminar delivered at ICM94 Santilli completed his Lie-admissible theory by showing that the algebra with Product (A, B) = AxRxB - BxSxA does indeed verify the Lie axioms (antisymmetry and Jacobi law), provided that the terms A<B and B>A are represented in their respective genoenvelopes <U and U>over corresponding genofields <F and .

To understand better how the Lie-admissible product (A, B) = AxRxB - BxSxA, with R different than S, can be antisymmetric, recall that conventional Lie algebra admit one single realization, that on conventional spaces and fields (read: with respect to the trivial unit I = Diag.(1, 1, ...1)); the isoalgebras admit instead a dual realization, that on isospaces over isofield (read: with respect to the isounit E) as well as the projection on conventional spaces over conventional fields (read: with respect to the conventional unit I); for the genoalgebras we have essentially the a similar occurrence, namely, they can be computed on the right and left genospaces over right and left genofields (read: right and left genounits E> and <E), in which case the product (A, B) verifies the Lie axioms, or it can be computed in its projection in conventional spaces and fields (read: with respect to the conventional unit I), in which case the product (A, B) is manifestly non-Lie.

Equivalently, the Lie character of the product (A, B) = A<B - B>A on genospaces over genofields can be seen from the fact that the lifting of the associative envelope AxB -> A>B = AxSxB is compensated by an INVERSE lifting of the unit I -> E> = 1/S, thus preserving the original structure (i.e., U and U> are isomorphic), and the same occurs for the right product (i.e., U and <U are also isomorphic). Thus, at the abstract, realization-free level, the product (A, B) verifies the anti-commutative law and the Jacobi law

(20a) (A, B){<S,S>} = -(B, A){<S, S>}

(20b) ((A, B), C){<S, S>} + ((B, C), A){<S, S>} + ((C, A), B){<S, S>} = 0.

PROPOSED RESEARCH V-4 : Study the Lie-admissible theory and its isodual with particular reference to: the genostructure theory, the transition from the genoalgebras to related genogroups; the representation theory; etc.

SIGNIFICANCE: Lie algebras are the algebraic counterpart of conventional geometries; Lie-Santilli isoalgebras are the algebraic counterpart of the isogeometries; and, along similar lines, genoalgebras are the algebraic counterpart of the genogeometries. The conventional, modular representation theory of Lie algebras characterize particles in linear, local, canonical and reversible conditions; the isomodular representation theory of Lie-Santilli isoalgebras characterize particles in nonlinear, nonlocal and noncanonical but still reversible conditions; the bi-modular representation theory of the genoalgebras characterizes particles in nonlinear, nonlocal, noncanonical as well as irreversible conditions, such as a neutron in the core of a neutron star. The most advanced definition of “particle” in physics, admitting all other as particular case, including those characterized by string and supersymmetric theories, is a bi-isomodular representation of the Lie-admissible covering of the Poincare’ symmetry.

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A further advance of the recent memoir [I-1] is that the genounits E> and <E can be ordered sets of nonhermitean quantities under ordinary operations, in which case the Lie-admissible theory becomes multivalued yet still preserving the original Lie axioms at the abstract level. A form of hyper-Lie theory defined via hyperoperations was also introduced by Santilli and Vougiouklis in ref. [II-2].

PROPOSED RESEARCH V-5: Study the multivalued realization of the Lie-admissible theory, first, with ordinary operations, and then with hyperoperations.

SIGNIFICANCE: Besides the evident mathematical significance, multi-valued spaces have already seen their appearance in cosmology, and their need in biology is now established in view of the complexity of biological systems.

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Prof. T. Vougiouklis
Department of Mathematics
Democritus University of Thrace
GR-67100 Xanthi, Greece, fax +30-551-39630

REFERENCES FOR SECT. V:

[V-1] R. M. Santilli, Foundations of Theoretical Mechanics, Vol. II, Springer-Verlag, 1983;
[V-2] D. S. Sourlas and Gr. Tsagas, Mathematical Foundations of the Lie-Santilli Theory, Ukraine Academy of Sciences, Kiev, 1993.
[V-3] J. V. Kadeisvili, An introduction to the Lie-Santilli isotheory, Rendiconti Circolo Matematico Palermo, Suppl. No. 42, pages 83-136, 1996.
[V-4] R. M. Santilli, Nonlinear, nonlocal and noncanonical, axiom-preserving isotopies of the Poincare’ symmetry, J. Moscow Phys. Soc. Vol. 3, pages 255-297, 1993.
[V-5] R. M. Santilli, Initiation of the representation theory of Lie-admissible algebras on a bimodular Hilbert space, Hadronic J. Vol. 3, pages 440-506, 1979.

RECENT PAPERS APPEARED IN THE RESEARCH PROPOSED OF SECT. V

STUDIES ON THE CLASSIFICATION OF LIE-SANTILLI ALGEBRAS
Gr. Tsagas
Department of Mathematics
Aristotle University
Thessaloniki 54006, Greece
Algebras, Groups and Geometries Vol., 13, pp. 129-148, 1996

AN INTRODUCTION TO THE LIE-SANTILLI NONLINEAR, NONLOCAL AND NONCANONICAL ISOTOPIC THEORY
J. V. kadeisvili
Institute for Basic Research
P. O. Box 1577
Palm Harbor, FL 34682 U.S.A. (br> ibr@gte.net
Mathematical Methods in applied sciences Vol. 19, pp.1349-1395, 1996

REMARKS ON THE LIE-SANTILLI BRACKETS
M. Nishioka Yamaguchi University
Yamaguchi 753, Japan
"New Frontiers in Algebras, Groups and Geometries",
Gr. Tsagas, Editor, Hadronic Press, Palm Harbor (1996), pp.553-558.

COMMENTS ON A RECENT NOTE BY MOROSI AND PIZZOCCHERO ON ON SANTILLI'S ISOTOPIES OPF LIE'S THEORY
J. V. kadeisvili
Institute for Basic Research
P. O. Box 1577
Palm Harbor, FL 34682 U.S.A. (br> ibr@gte.net
Submitted for ppublication

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