sign in sign up
<<
Home , Malcev algebra, Moufang loop, Octonion, Power associativity

Split

Benjamin Prather

Florida State University Department of

November 15, 2017

Split Octonions Let R be a ring (with unity). Prather Let G be a group. Loop Algebras

Octonions Denition Moufang Loops An element of group R[G] is the formal sum: Split-Octonions Analysis X Malcev Algebras rngn

Summary gn∈G Addition is component wise (as a free module). follows from the products in R and G, distributivity and commutativity between R and G.

Note: If G is innite only nitely many ri are non-zero. Group Algebras

Split Octonions

Prather

Loop Algebras A group algebra is itself a ring. Octonions In particular, a group under multiplication. Moufang Loops

Split-Octonions Analysis A set M with a is called a . Malcev Algebras Summary This process can be generalized to magmas. The resulting algebra often inherit the properties of M. Divisibility is a notable exception. Magmas

Split Octonions

Prather

Loop Algebras

Octonions

Moufang Loops

Split-Octonions

Analysis

Malcev Algebras

Summary Loops

Split Octonions

Prather

Loop Algebras In particular, loops are not associative. Octonions It is useful to dene some weaker properties. Moufang Loops power associative: the sub-algebra generated by Split-Octonions

Analysis any one element is associative. Malcev Algebras diassociative: the sub-algebra generated by any Summary two elements is associative. associative: the sub-algebra generated by any three elements is associative. Loops

Split Octonions

Prather

Loop Algebras Octonions gives us (xx)x = x(xx). Moufang Loops This allows x n to be well dened. Split-Octonions

Analysis −1 Malcev Algebras Diassociative loops have two sided inverses, x . −1 −1 Summary Diassociative gives us (xy)x = x(yx ). This allows conjugation to be well dened. Loop Algebras

Split Octonions

Prather Let L be a loop. Loop Algebras Denition Octonions An element of loop algebra is the formal sum: Moufang Loops R[L] Split-Octonions X Analysis rnln Malcev Algebras ln∈L Summary Addition is component wise (as a free module). Multiplication follows from the products in R and L, distributivity and commutativity between R and L.

Note: If L is innite only nitely many rn are non-zero. loop

Split Octonions Let be the loop dened by this table. Prather O

Loop Algebras e0 e1 e2 e3 e4 e5 Octonions e1 −e0 e3 −e2 e5 −e4 −e7 e6 Moufang Loops

Split-Octonions e2 −e3 −e0 e1 e6 e7 −e4 −e5

Analysis e3 e2 −e1 −e0 e7 −e6 e5 −e4 Malcev Algebras e4 −e5 −e6 −e7 −e0 e1 e2 e3 Summary e5 e4 −e7 e6 −e1 −e0 −e3 e2 e6 e7 e4 −e5 −e2 e3 −e0 −e1 e7 −e6 e5 e4 −e3 −e2 e1 −e0

Note: There are really 16 elements, ± for each shown. 2 Formally, −en is really en, where  = 1 and commutes. Other Representations

Split Octonions

Prather

Loop Algebras Let e0 be the identity and in be indexed mod 7. Let 2 and ( ). Octonions in = −e0 inim = −imin m 6= n Moufang Loops Let inin+1 = in+3. This is isomorphic to O. Split-Octonions

Analysis One such mapping is: 1 1, 2 2, 3 4. Malcev Algebras i = e i = e i = e This forces , , , . Summary i4 = e3 i5 = e6 i6 = −e7 i7 = e5 Also in+1in+3 = in and in+3in = in+1.

This representation exhibits a seven fold symmetry in O. Other Representations

Split Octonions

Prather

Loop Algebras 7 lines and 7 points.

Octonions Central circle is a line.

Moufang Loops Split-Octonions Three points on each line. Analysis Three lines at each point. Malcev Algebras

Summary Points labeled with roots. Lines labeled to indicate positive products. Other Representations

Split Octonions

Prather

Loop Algebras

Octonions

Moufang Loops

Split-Octonions

Analysis

Malcev Algebras

Summary

John H. Conway Gino Fano Octonion Algebras

Split Octonions

Prather

Loop Algebras (e1e2)e4 = e3e4 = e7 = −e1e6 = −e1(e2e4). Octonions Thus O is not associative. Moufang Loops

Split-Octonions

Analysis Also note that any two elements generate

Malcev Algebras the associative group. Summary Thus O is diassociative.

The anti-, q, toggles the of e1 to e7. Octonion Algebras

Split Octonions

Prather Let I = h1 + i if 1 6= −1 or h1i otherwise. Loop Algebras This formally associates  with −1 in R, Octonions unless 2. Moufang Loops char(R) = is the loop algebra . is . Split-Octonions OR R[O]/I O OR Analysis Malcev Algebras N(q) = qq = qq Summary These are composition algebras, N(xy) = N(x)N(y).

Over R only R, C, H and O are positive denite composition algebras. Discovery of the Octonions

Split Octonions

Prather

Loop Algebras

Octonions

Moufang Loops

Split-Octonions

Analysis

Malcev Algebras

Summary William R. John T. Graves Aurther Cayley Hamilton Algebras

Split Octonions

Prather Theorem (Moufang's theorem) Loop Algebras Let an Loop L satisfy Octonions Moufang Loops xy · zx = x · yz · x Split-Octonions x · y · xz = xyx · z Analysis

Malcev Algebras xy · z · y = x · yzy Summary Let x, y and z be elements of L. Ruth If x(yz) = (xy)z, then x, y and z Moufang generate an associative sub-loop (i.e a group).

Loops with these properties are Moufang loops. Moufang Loops

Split Octonions

Prather

Loop Algebras All Moufang loops are diassociative. Octonions Thus the RHS are well dened. Moufang Loops

Split-Octonions In particular, the units of are Moufang loops. Analysis OR Malcev Algebras Summary If K is a eld we can dene q−1 = q/N(q). This makes a . OK Projective spaces

Split Octonions

Prather Theorem (Desargues' theorem) Loop Algebras

Octonions Two triangles are in perspective axially i they are in

Moufang Loops perspective centrally.

Split-Octonions

Analysis This was proved for Euclidean spaces. Malcev Algebras It applies to all projective spaces, if dim 6= 2. Summary Hilbert found counter examples when dim = 2. Moufang showed that if an algebra is non-associative it's projective space is non-Desargues. In particular, she was interested in Cayley Planes. Projective spaces

Split Octonions Prather Thus there is a non-Desarguesian octonionic projective Loop Algebras plane, but no higher octonionic projective spaces. Octonions

Moufang Loops

Split-Octonions

Analysis

Malcev Algebras

Summary

Girard Desargues David Hilbert Parker Loop

Split Octonions Prather The Golay code is a 12 dimensional linear code 24 Loop Algebras over with a natural representation in . F2 F2 Octonions They can be viewed as sets of bits. Moufang Loops The Parker Loop is a created as Split-Octonions a double of the Golay Code, by adding a sign bit. Analysis

Malcev Algebras Summary Let HW (x) be the cardinatity of code point x. x 2 = (−1)HW (x)/4 xy = (−1)HW (x∩y)/2yx x(yz) = (−1)HW (x∩y∩z)(xy)z Note: The various sign choices here are isomorphic. Parker Loop

Split Octonions Prather Parker showed that this is indeed a Moufang Loop.

Loop Algebras Octonions Conway used this loop to simplify Griess' construction Moufang Loops of the Fischer-Griess . Split-Octonions (Griess's constructed it as the of a Analysis

Malcev Algebras commutative, non-associative, algebra).

Summary The monster group plays a signicant role in the classication of nite simple groups. So Moufang loops play a role in one of the biggest, if not most important, theorems of modern mathematics. Parker Loop

Split Octonions

Prather

Loop Algebras

Octonions

Moufang Loops

Split-Octonions

Analysis

Malcev Algebras

Summary Marcel J. E. Richard A. Parker Robert Griess Golay Paige Loops

Split Octonions

Prather Moufang loop are diassociative.

Loop Algebras Thus conjugation is well dened.

Octonions Moufang Loops This allows us to dene a normal sub-loop, Split-Octonions and simple Moufang loops in the usual way. Analysis

Malcev Algebras Summary Paige showed that the units of split-octonion algebras over a nite eld are simple. Liebeck showed that their are no other cases.

Finite simple Moufang loops are called Paige loops. Paige Loops

Split Octonions

Prather

Loop Algebras

Octonions

Moufang Loops

Split-Octonions

Analysis

Malcev Algebras

Summary

Lowell Paige Martin Liebeck Split-Octonions

Split Octonions

Prather Consider . OC Loop Algebras Clearly the real parts of this algebra form O. Octonions However, it contains another algebra over . Moufang Loops √ R Specically, multiply each , 4 by 1. Split-Octonions en n ≥ −

Analysis Malcev Algebras The resulting algebra has a split signature. Summary Thus they are called the split-octonions, O−.

Rename as for 0 4, since 2 1. √ en in < n < in = − Also, 1 as for 4, since 2 1. − en jn n ≥ jn = Split-Octonions

Split Octonions

Prather 2 2 2 2 2 2 2 2 Loop Algebras N(q) = qq = r0 + r1 + r2 + r3 − r4 − r5 − r6 − r7 − Octonions O not positive denite, as N(j4) = −1. Moufang Loops Split-Octonions − Infact, O has zero divisors, N(1 + j4) = 0. Analysis The zero divisors all have 0. Malcev Algebras N(q) =

Summary O− has the composition property. The elements with N(q) 6= 0 are invertible. The invertible elements form a loop. Split Sub-Algebras

Split Octonions

Prather

Loop Algebras q is a split root if N(q) = −1 (proper if N(q) = 1).

Octonions Moufang Loops Any split root denes a split-complex algebra C−. Split-Octonions Any two orthogonal split roots dene a Analysis split-quaternion algebra, −. Malcev Algebras H

Summary Any sub-algebra is either one of these, C or H. In particular, the algebra generated by a proper root and an orthogonal split root is also H−. Lorentz Transforms

Split Octonions

Prather − ∼ C = R × R. This algebra describes the Loop Algebras Lorentz transformations in one . Octonions

Moufang Loops − ∼ Split-Octonions H = M2(R). This algebra describes the Analysis Lorentz transformations in two . Malcev Algebras The third basis, N(i3) = 1, models Summary Thomas precession.

Can we use O− to model Lorentz transformations in three dimensions? Cl(0, 3) is more standard. Vector Representation

Split Octonions Let J~ be three split generators, Prather interpreted as an axial vector. Loop Algebras Let ~I represent a polar vector. Octonions Let k represent a pseudo scalar. Moufang Loops Up to a sign convention on k we get the following table: Split-Octonions Analysis ~ ~ Malcev Algebras e J I k Summary · : e · : −k J~ −~I × : ~I × : −J~ · : k · : −e ~I −J~ × : −J~ × : ~I k ~I J~ e Zorn Representation

Split Octonions

Prather These representations of C− and H− Loop Algebras use idempotents as a basis. Octonions This halves the cost of multiplication. Moufang Loops

Split-Octonions Analysis Zorn found a similar representation for O−. Malcev Algebras Let a = e0 + j4, b = e0 − j4. Summary Max Zorn Let I = a hi1, i2, i3i and J = −b hi1, i2, i3i. a I  Known for q = his lemma J b N(q) = det(q) = ab − J · I Zorn Matrix Representation

Split Octonions This product is then −. Prather O

Loop Algebras q00 = qq0 Octonions 00 0 0 Moufang Loops a = aa + I · J Split-Octonions I 00 = aI 0 + Ib0 + J × J Analysis 00 0 0 Malcev Algebras J = Ja + bJ − I × I 00 0 0 Summary b = J · I + bb

This is just matrix multiplication, plus two cross products. Note: There are several sign conventions possible, impacting only the terms. Fueter Analysis

Split Octonions

Prather In 1936 Fueter developed the theory of

Loop Algebras analysis on the .

Octonions Moufang Loops The dierence operator is too strict, Split-Octonions only linear functions have a derivative. Analysis

Malcev Algebras Summary Taylor series are too loose. Functions exist that converge on domains, but are not what we intuitively feel are smooth.

Generalizing the concept of analytic is the just right. Fueter Analysis

Split Octonions The quaternions are not commutative, Prather so we have to dene left and right analytic. Loop Algebras

Octonions

Moufang Loops This uses The following Dirac operators: P ∂ P ∂ Split-Octonions D = ei D = ei i ∂ri i ∂ri Analysis Malcev Algebras A smooth function from a domain in 8 Summary R to the octonions is left (right) analytic if: P ∂f P ∂f Df = ei = 0 (fD = ei ) = 0 i ∂ri i ∂ri

Note that DD = DD is the Laplacian operator. Thus analytic functions are harmonic functions. Fueter Analysis

Split Octonions

Prather

4 Loop Algebras Next we view the quaternions as R , Octonions and generate a Cliord algebra over 4. Moufang Loops R

Split-Octonions

Analysis A quaterion cooresponds to a 1-form, and its exterior

Malcev Algebras derivative cooresponds to a 3 form  which has precisely Summary a quaternionic value!

Denote these dr and ?dr, where ? is the Hodge duel. Fueter Analysis

Split Octonions

Prather

Loop Algebras

Octonions

Moufang Loops

Split-Octonions

Analysis

Malcev Algebras

Summary Karl Rudolf Brook Taylor Paul Dirac Fueter Fueter Analysis

Split Octonions

Prather Generalizing the Cauchy Kernel, r , Φ(r) = N(r)2 Loop Algebras plus a bit of integral gymnastics Octonions

Moufang Loops gives us a version of the Cauchy integral formula. 1 R , Split-Octonions f (r0) = 2π2 ∂U Φ(r − ro) · ?dr · f (r) Analysis with the usual domain restrictions. Malcev Algebras Summary From here it is down hill to generalize many of the theorems of complex analysis, in particular: The mean value, maximum modulus and uniform convergence theorems. Octonionic Analysis

Split Octonions

Prather

Loop Algebras In 2002 Li and Peng generalized this process to the Octonions octonions. Here associativity comes into play. Moufang Loops r Φ(r) = 4 , Split-Octonions N(r) 3 R , Analysis f (r0) = π4 ∂U Φ(r − ro) · (?dr · f (r)) Malcev Algebras Summary In particular, 3 R  π4 ∂U (Φ(r − ro) · ?dr) · f (r) − f (r0) R P 0. = U i [Φ, Dfi , ei ]dV 6= Octonionic Analysis

Split Octonions

Prather

Loop Algebras

Octonions

Moufang Loops

Split-Octonions

Analysis

Malcev Algebras

Summary Augustin-Louis Xingmin Li Lizhong Peng Cauchy Octonionic Analysis

Split Octonions

Prather In 2011 Libine generalized this to the split-quaternion. At zero divisors the Cauchy Kernel becomes zero. Loop Algebras The solution is to complexify the algebra and perturbe it. Octonions r Moufang Loops Φ(r) = (N(r)+i|r|2)2 Split-Octonions If the boundary of our domain crosses the null cone Analysis transversely, the following limit converges. Malcev Algebras Summary −1 Z f (r0) = lim Φ(r − r ) · ?dr · f (r) 0 2 2 o → π ∂U

Any convex domain with the point in the interior will satisfy this. In particular cubes and spheres. Octonionic Analysis

Split Octonions

Prather

Loop Algebras

Octonions

Moufang Loops

Split-Octonions

Analysis

Malcev Algebras

Summary

Matvei Libine Split-Octonionic Analysis?

Split Octonions

Prather Using these two outlines, this very same process Loop Algebras should work for the split octonions! Octonions Deriving the appropriate Cauchy integral formula Moufang Loops is the rst step in this investigation. Split-Octonions

Analysis Malcev Algebras Perhaps something like: Summary r Φ(r) = (N(r)+i|r|2)4 −3 Z f (r0) = lim Φ(r − r ) · (?dr · f (r)) 0 4 o → π ∂U Malcev Algebras

Split Octonions

Prather Malcev studies whether one can generalize Lie algebras Loop Algebras in group theory to similar algebras over Moufang loops. Octonions

Moufang Loops

Split-Octonions For example, the tangent space of the identity of Analysis a smooth Moufang loop forms a . Malcev Algebras Summary 7 8 S , interpreted as the unit octonions in R is a quintisential example of a smooth Moufang loop. The resulting Malcev algebra is equivelent to the pure octonions with [x, y] = xy − yx. Malcev Algebras

Split Octonions

Prather Using the split octonions with N(x) = 1 we get Loop Algebras a smooth Moufang loop on a hyperboloid. Octonions The resulting Malcev algebra is equivelent to the Moufang Loops

Split-Octonions pure split-octonions with the same product.

Analysis Malcev Algebras If a Moufang loop is a connected, simply connected Summary and real-analytic, it can be recovered from its Malcev algebra.

Some formula need to be modied. Malcev Algebras

Split Octonions

Prather

Loop Algebras J(x, y, z) = [x, [y, z]] + [y, [z, x]] + [z, [x, y]] 6= 0 but Octonions [J(x, y, z), x] = J(x, y, [x, z]). Moufang Loops

Split-Octonions Let Lx and Rx be the action of left and right Analysis multiplication by x. Malcev Algebras 2 0 Summary [Lx, Ly] − L[x,y] = − [Lx , Ry ] = [Rx , Ry ] + R[x,y] 6=

In 2006 Eugen Paal demonstrated that one can dene Noether currents and charges whithin this framework. Malcev Algebras

Split Octonions

Prather

Loop Algebras

Octonions

Moufang Loops

Split-Octonions

Analysis

Malcev Algebras

Summary Anatoly Ivanovich Sophus Lie Eugen Paal Malcev Summary

Split Octonions Prather Octonions are neat!

Loop Algebras Split octonions might be neater! Octonions They are related to many exceptional objects. Moufang Loops Split-Octonions Much of uses associativity Analysis

Malcev Algebras to get to a well dened conjugation.

Summary This machinery applies to Moufang loops. These tools then apply to create exceptional objects.

They break when pushed too far, when associativity is strictly needed.