Algebraic Loops (2); Symmetry-Conserving Vector-Division Hoop Algebras

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An algebraic loop with the Moufang property, for all , , in the loop, acts as a vector multiplication and division table for unsigned vectors. If and , then ; the left-multiplicative inverse is .


A Moufang loop is alternative, , and all groups, which are associative, , are Moufang loops. If a Moufang loop has elements and -fold symmetry it can be "folded" to a loop algebra with -signed vectors having () elements. This introduces roots of unity as generalized signs. Real and complex algebras arise when or ; examples with are also included here. Unsigned loops operate as algebras over real and complex fields because they can be folded from larger loops with C2 or C4 symmetries. Loop algebras may have divisors of zero, and include non-Abelian (noncommutative) and nonassociative cases.

I define hoops as symmetry-conserving loop algebras, in which the conjugate factors of the table determinant are conserved sizes or symmetries because (up to a sign). All groups conserve their determinants (Frobenius 1895); some nonassociative (octonionic) loops are also shown here to be hoops. Hoops lead to new mathematical concepts that are developed in this set of Demonstrations—unsigned continuous "primal" numbers (the half-line) are fundamental, generalized signs are created by folding Moufang loops; the "real algebras without divisors of zero" {, , , } are degenerate mono-sized hoops; vector division-by-zero is eliminated by the creation of symmetry-conserving remainders; zero sizes project results into reduced-symmetry sub-hoops; Abelian hoops have multi-angle polar-duals and multiphase sinusoidal orbits; etc. The Clifford(,) anti-commutative algebras are an important subset of hoops; they generalize both geometry and analysis to any number of dimensions; they have elements that are square roots of unity and that are fourth roots, together with a pseudoscalar "I" that is an anti-commutative version of , so neither -1 nor are unique.

Choose a "name" (there are 73 available, including 46 distinct, non-isomorphic, hoops). The multiplication table (shown in index or symbolic form) and a brief description appear. Some short sizes are shown. If the table is a group, gmmnn is shown—the group is the entry in the list of groups with mm elements in the GAP group atlas. Two "random" vectors and their product will also appear. Now step some elements (the values wrap round from +9 to -9). The product changes. The shapes (lists of sizes or symmetries) also appear. If , the table is reported to be conservative.

Some nonassociative Moufang loops, with up to 16 elements and in their names, are provided. Only a few are conservative. A few counterexamples, with names ending in or , are nonconservative loops.

Choose a hoop with several sizes. Adjust and until they have different zero sizes. The sizes of will have both sets of zeros. Hoop algebras conserve these lost symmetries by ejecting "remainders", , just as particle interactions may create several particles that conserve symmetries.

Hoops have multiplicative inverses , with (the unit vector). This is demonstrated by recovering from . splits into partial fractions with sizes as denominators. As zero sizes in multiplicands give the same zeros in the product, they project the result into a subalgebra of reduced symmetry, ejecting remainders if necessary. The zero is factored out of the calculations, and partial fractions with denominators of near zero size are excluded from the inverse. This eliminates division by zero in hoop operations on vectors.

Later Demonstrations show Abelian (commutative) hoops with polar dual formulations that provide powers and roots, and anticommutative Clifford (geometric) algebras that unify many areas of mathematics.


Contributed by: Roger Beresford (March 2011)
Open content licensed under CC BY-NC-SA



Algebraic loop Demonstrations are based on the GroupLoopHoop package in MathSource/4894, which contains much more information. The first develops loop properties and the third describes small groups and Moufang loops. Demonstrations of Abelian loops with Cartesian/polar duals, and Clifford (geometric) algebras (which are anticommutative hoops) are being prepared.

Integer elements are used for compactness, but real and complex numbers behave similarly. Output is real, to avoid bulky results involving square-root symbols.

Multiplication . If the table has in row and column , then the element of the product is the sum of A[[i]] B[[j]] Sign[k]. If the product has zero sizes where or has a nonzero size (because and have disparate zero sizes), nonzero remainders are created with the missing size(s). The left remainder is and the right remainder is . Compare this with integer division with remainder and , so recovers . As every vector has an inverse, division is simply multiplication by an inverse.

Inverse . If the shape of has symbolic form S[[k]] (a list of polynomials) and the table has a 1 (possibly signed) in row of column (this data is supplied in ), then the element of the (left) inverse is the signed sum of the derivations of S[[k] with respect to symbolic variable[[n]], divided by S[[k]]. This partial fraction is omitted if S[[k]] is zero; the zero has been factored out of the determinant.

An outstanding research problem is the development of compact shape expressions for some tables (some are shown with "{0}" in their shapes); division cannot be implemented in such cases.

The discovery that hoops (including all Clifford algebras) have exactly conserved sizes (and not just norms with triangle inequalities) appears to be new.

Hoops include compact descriptions of all the polar-complex and planar-complex analytic algebras developed (at great length, but without shapes) in Sylviu Olariu's "Complex Numbers in n Dimensions".

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