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Algebraic Loops, Groups, "Hoops" and Clifford algebras.

Roger Beresford
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ALGHCA.cdf is a package of procedures to create, manipulate, identify, and use small Algebraic Loops, Groups, "Hoops" and Clifford algebras. Hoops are relevant to physics because their Moufang vector-division and Frobenius conservation properties provide the conserved symmetries that define forces and particles. A database includes over 80 hoops and most groups with up to 96 elements. ALGHACAtest.cdf validates the database, ALGHCAGloss.cdf explains the terms employed, Hoops.cdf explains the procedures and provides example calculations. The notebooks are designed to be read by non-Mathematica users who have downloaded CDFPlayer; Mathematica users (Version 7 or later) can test them with their own data. They provide background material on algebraic loops and groups and then demonstrate "Symmetry-conserving Generalised-sign Vector-division Algebras" or "Hoops". Hoops unify most standard algebras - complex, Cayley-Dickson, Clifford, Davenport, Dirac, non-commutative, octonion, Olariu, Pauli-sigma, quaternion, real, spinor, wedge, etc. algebras. A database of over 900 loops is provided. Many "mathematical truths" are only valid in the context of real and complex numbers because they are degenerate cases of more general "Hoop" relationships.

The key features of Hoop algebras are:

1) Algebraic Loops define multiplication of "vecs" (sets of unsigned "Primal" numbers). Loops having Cr (the cyclic group with r elements) as a central subgroup can be "folded" to introduce r'th roots of unity as "generalised signs".

2) "Folding" converts loops (with one operation, multiply) into algebras with generalised addition/subtraction as a second operation; vecs fold to vectors. Real and complex numbers are folded from pairs (r=2) or quads (r=4) of primal numbers.

3) Hoops are algebras possessing Frobenius's determinant-conservation property. This provides symmetries (or "sizes", factors of the symbolic multiplication table) that are conserved on multiplication. Sizes are also denominators for the partial-fraction formulation of multiplicative inverses.

4) Multiplication and division are implemented by the "hoopTimes" procedure; division is pre-multiplication by the inverse, given by "hoopInverse".

5) Sizes can be zero; operations are then "projected" onto a sub-algebra (of reduced symmetry) with the same sizes zeroed. To maintain size conservation, hoopTimes "projects" results into the sub-algebra and "ejects" remainders. This resembles particle interactions and decays, with remainders corresponding to ejected particles with different symmetries.

6) Some hoops have Polar-duals with additive angles, generalizing the x,y and r,theta complex-plane duality. This allows the calculation of vector powers and roots, via "hoopPower".

7) Some unital Polar-duals have (deBroglie-like) multi-phase orbits and intrinsic (Planck-like) radii. Their elements include half-spin quantum operators that are neither real nor complex. Hoop algebras give tantalizing suggestions about the re-interpretation of mathematical physics - they may describe bosons and fermions. Ternary (quark) symmetries occur without invoking octonion triality. Kaluza-Klein orbital velocities could provide mass. Unital orbits may explain the "law of large numbers". Hoop maths may describe non-point-like particles with an intrinsic sub-Planck scale.

Complex numbers can express much of mathematics, but are a Procrustean bed; some parts relevant to physics do not fit on it. I have scratched the surface of a more fundamental mathematics, and found many features that may be relevant to mathematical physics. 8 May 2012.

*Science > Physics

Moufang loops, conservative algebras, algebras, Cayley-Dickson, Cayley-tables, Clifford, Clifford-like, conservative loops & algebras, continuous orbits, determinants, directors, Frobenius-conservation, generalized signs, groups, graded algebras, hoops, loops, isomers, isomorphs, m-vecs, plex-conjugate, Primal numbers, Roger algebras, shapes & sizes (of vecs), signed-tables, partial-division-by-zero, quasigroups, renormalization, roots of unity, vecs, univectors.
Related items

*Groups, Loops, & Hoop Algebras   [in MathSource: Packages and Programs]

ALGHCA.cdf (2.1 MB) - CDF Document
ALGHCAGloss.cdf (102.3 KB) - CDF Document
ALGHCAtest.cdf (677.7 KB) - CDF Document
Hoops.cdf (1.7 MB) - CDF Document
frontpage.cdf (7.6 KB) - CDF Document