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Groups, Loops, & Hoop Algebras

Roger Beresford
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GroupLoopHoop.m is a package of procedures to create, manipulate, identify, and use small Groups, Algebraic Loops, and "Hoop 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 all groups with up to 72 elements. GroupLoopTest.nb validates the database, HoopGlossary.nb explains the terms employed, GroupLoopDemo.nb and HoopDemo.nb demonstrate the procedures. MathSource/6198 is a much smaller and simpler database and explanation, restricted to Hoop algbras and omitting the background material on Groups & Loops. The simplifications make it incompatible with this notebook.

The notebooks are designed to be read by, and comprehensible to, non-Mathematica users who have downloaded MathReader; Mathematica users (Version 4.2 or later) can test them with their own data. They provide background material on algebraic loop and groups and then demonstrate "Conservative Partial-fraction-division Algebras" or "Hoops". Hoops unify most standard algebras - Complex, Cayley-Dickson, Clifford, Davenport, Dirac, non-commutative, Octonion, Pauli-sigma, Quaternion, Real, Octonion, Spinor, Wedge, etc., together with new "terplex" algebras. As all groups and octonions are Moufang Loops, a database of over 800 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: Moufang Loops define multiplication and division of "directors" (sets of unsigned "Primal" numbers). The Moufang division property ensures that every director has a multiplicative inverse. Generalised signs are r'th roots of unity; r-fold negations are equivalence relations that "fold" m×m Moufang Loops (if they have r-fold symmetry) to (m/r)×(m/r) multiplication tables for Algebras. Directors fold to generalized vectors or "vecs". Real and complex numbers are folded from pairs (r=2) quads (r=4) of primal numbers. Folding involve a loss of information. Hoops are defined as algebras with "sizes" (symmetries, factors of the symbolic multiplication table) that are conserved on operations with vecs. They are also denominators for the partial-fraction formulation of multiplicative inverses. Multiplication and division are implemented by the "hoopTimes" procedure; division is pre-multiplication by the inverse, given by "hoopInverse".

Vec 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 may "project" results into the sub-algebra and "eject" remainders. This resembles particle interactions and decays, with remainders corresponding to ejected particles with different symmetries. 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". Some unital Polar-duals have (deBroglie-like) multi-phase orbits and intrinsic (Planck-like) areas. 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, which (at 77) I cannot expect to explore. 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 underlie (and simplify) "M-theory", describing non-point-like particles with an intrinsic 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, but "Non omnia possumus omnes" (Virgil).

Version 15.47. 2 May 2012.

*Mathematics > Algebra > Group Theory
*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

*Hoop Algebras   [in MathSource: Packages and Programs]

Downloads Download Wolfram CDF Player

ALGHCA.nb (2.1 MB) - AlgebraicLoopGroupHoop Package
ALGHCAGloss.nb (88.5 KB) - Glossary for Loops, Groups, Hoops, Cliffords
ALGHCAtest.nb (447.9 KB) - Validation of ALGHCA.nb
BeresfordV5.zip (575.1 KB) - Zip of all V5 notebooks [for Mathematica 5.2]
BeresfordV6.zip (862.4 KB) - Zip of all V6 notebooks [for Mathematica 6.0]
GroupLoopDemo.nb (1.2 MB) - Mathematica Notebook [for Mathematica 5.2]
GroupLoopHoop.m (480.6 KB) - Mathematica Package [for Mathematica 5.2]
GroupLoopHoop.nb (687.1 KB) - Mathematica Notebook [for Mathematica 5.2]
GroupLoopTest.nb (136.2 KB) - Mathematica Notebook [for Mathematica 5.2]
HoopDemo.nb (508 KB) - Mathematica Notebook [for Mathematica 5.2]
HoopGlossary.nb (394.7 KB) - Mathematica Notebook [for Mathematica 5.2]
Hoops.cdf (1.7 MB) - Hoops CDF