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Split Octonions Split Octonions Benjamin Prather Florida State University Department of Mathematics November 15, 2017 Group Algebras 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 algebra R[G] is the formal sum: Split-Octonions Analysis X Malcev Algebras rngn Summary gn2G Addition is component wise (as a free module). Multiplication 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 binary operation is called a magma. 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 Power associativity 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 ln2L 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. Octonion loop Split Octonions Let be the loop dened by this table. Prather O Loop Algebras e0 e1 e2 e3 e4 e5 e6 e7 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. Fano Plane 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 quaternion group. Summary Thus O is diassociative. The anti-automorphism, q, toggles the sign 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 Division 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 division algebra. 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 Moufang loop 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 Monster group. Split-Octonions (Griess's constructed it as the automorphism group 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 p 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. p 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 dimension. Octonions Moufang Loops − ∼ Split-Octonions H = M2(R).
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