Sedenions can be considered to be a ZeroDivisor Algebra. Consider the following theorems (stated here without proof):

Generalized Frobenius: The alternative real division algebras, alternativity being defined in terms of the associator (x,y,z) = x(yz) - (xy)z by (x,y,z) = (-1)^P (Px,Py,Pz) for permutation P of x,y,z are: real numbers R; complex numbers C; quaternions Q; and octonions O. In q-alg/9710013 Guillermo Moreno describes The Zero Divisors of the Cayley-Dickson Algebras over the Real Numbers. He shows, among other things, that the set of zero divisors (with entries of norm one) in the sedenions is homeomorphic to the Lie group G2. If the sedenions are regarded as the Cayley-Dickson product of two octonion spaces, then: if you take one 7-sphere S7 in each octonion space, and if you take G2 as the space of zero divisors, then YOU CAN CONSTRUCT FROM THE SEDENIONS the Lie group Spin(0,8) as the twisted fibration product S7 x S7 x G2. Such a structure is represented by the design of the Temple of Luxor.

Robert de Marrais has constructed detailed models for Zero Divisors of algebras of dimension 2^N, with explicit results through N = 8. He ( see his paper math.GM/0011260 ) visualizes the zero divisors of sedenions and higher-dimensional algebras in terms of Singularity / Catastrophe Theory, saying "... 168 elements provide the exact count of "primitive" unit zero-divisors in the Sedenions. ( [there are] Composite zero-divisors, comprising all points of certain hyperplanes of up to 4 dimensions ... ) The 168 are arranged in point-set quartets along the "42 Assessors" [in the Egyptian Book of the Dead the soul of the deceased passes through a hall lined by 21 pairs of Assessors] (pairs of diagonals in planes spanned by pairs of pure imaginaries, each of which zero-divides only one such diagonal of any partner Assessor). Wave-interference dynamics, depictable in the simplest case by Lissajous figures, ... derive from mutually zero-dividing trios of Assessors, a D4-suggestive 28 in number ... ". He also constructs "box-kites" that are similar to Onarhedra/heptavertons. Although he does not prove it in his paper, he conjectures that his box-kites are related to the H3 Coxeter Group Singularity and, that it, in turn, is related to the 16-dimensional Barnes-Wall lattice. I speculate that the H4 Coxeter Group Singularity, being based on the symmetries of the 600-cell, which in turn can be constructed from a 24-cell, may be related to the 24-dimensional Leech lattice.

Hurwitz: The normed composition algebras with unit are: real numbers; complex numbers; quaternions; and octonions. Dixon: The algebras generated by sequences that are both Galois sequences and quadratic residue codes over GF(2) are: real numbers - GF(1) and code length 0; complex numbers - GF(2) and code length 1; quaternions - GF(4) and code length 3; and octonions - GF(8) and code length 7. Bott-Kervaire-Milnor: (see Numbers, by Ebbinghaus, Hermes, Hirzebruch, Koecher, Mainzer, Neukirch, Prestel, and Remmert, Springer-Verlag 1991, including Chapter 11, Division Algebras and Topology, by F. Hirzebruch; and Battelle Rencontres, ed. by Cecile M. DeWitt and John A. Wheeler, W. A. Benjamin 1968, including Chapter XIX, Continuous Solutions of Linear Equations - Some Exceptional Dimensions in Topology, by Beno Eckmann) By a theorem of Frobenius (1877), there are three and only three associative finite division algebras: 1-dimensional real numbers R; 2-dimensional complex numbers C; and 4-dimensional quaternions Q. By a theorem of Zorn (1933), every alternative, quadratic, real non-associative algebra without divisors of zero is isomorphic to the 8-dimensional octonions O. In his paper, Fraleighs misstag: en varning, Ernst Dieterich gives an example of a 4-dimensional real division algebra that is NOT R, C, Q, or O, even though it does have an identity 1 and a multiplicative inverse. It is the vector space R x R3 with the product (a,v)(b,w) = ( ab - v*w + v* Bx w , aw + bv + (By + Dd)(v x w) ) where * is transpose; a,b in R; v,w in R3; and (x,y,d) in K; K = R3 x R3 x T; T = {d in R3| 0<d1<d2<d3}; and 0 -x3 x2 d1 0 0 B = x3 0 -x1 Dd = 0 d2 0 (real entries). -x2 x1 0 0 0 d3 The value K = (0,0,(1,1,1)) gives the quaternions Q, but for other values of the 9 parameters, it is not Q. All members of the 9-parameter family are power-associative, that is every subalgebra generated by 1 element is associative, but only the quaternion algebra Q is alternative, that is every subalgebra generated by 3 elements is associative (in the case of Q, subalgebras generated by 3 independent elements generate Q itself, so Q itself is also associative). In 2 dimensions, every power-asociative real division algebra is isomorphic to the complex numbers C. In 8 dimensions, a construction similar to the one given above for dimension 4, but for R x R7 using 7-dimensional vectors and 7x7 antisymmetic matrices gives a 21+21+7 = 49-parameter family of 8-dimensional power-associative algebras that includes as one of its members the alternative octonions O. (You can use a Swedish Schoolnet web page to translate from Swedish to English.) (My thanks to Daniel Asimov for telling me about Ernst Dieterich's work.) In 1958, Kervaire and Milnor proved independently of each other that the finite-dimensional real division algebras have dimensions 1, 2, 4, or 8. To do so, they used the periodicity theorem of Bott on the homotopy groups of unitary and orthogonal groups. In 1960, Adams proved that a continuous multiplication in R(n+1) with two-sided unit and with norm product exists only for n+1 = 1, 2, 4, or 8. Ernst Dieterich says, in an abstract of a talk on Real Quadratic Division Algebras: "... [ R,C,Q,O ]... classifies all alternative real division algebras. A natural further generalization to ask for is the classification of all quadratic real division algebras. Due to the famous (1,2,4,8)-theorem ... it suffices to classify these in dimensions 4 and 8. ... Making systematic use of the basic general fact that a quadratic algebra is `the same thing as' an anti-commutative algebra endowed with a bilinear form (Osborn 1962), we shall derive both a complete classification of all 4-dimensional real quadratic division algebras and the construction of a large family of 8-dimensional real quadratic division algebras which is conjectured to come close to a classification. ...", and in a paper Eight-Dimensional Real Quadratic Division Algebras: "... A real algebra A is called quadratic in case 0 < dim A, there exists an identity element 1 in A and each x in A satisfies an equation x^2 = ax + b1 with real coefficients a,b. Every alternative real division algebra is quadratic. The classification of all quadratic real division algebras seems to be within reach. It is based on their intimate relationship with dissident triples ...". Examples of dissident triples are: space R3, map Bx: R3/\R3 -> R,and dissident map (By+Dd): R3/\R3 -> R3 that gives the 9-parameter family of 4-dimensional power-associative algebras; and space R7, map Bx: R7/\R7 -> R,and dissident map (By+Dd): R7/\R7 -> R7 that gives the 49-parameter family of 8-dimensional power-associative algebras. As Dave Rusin says on his web site: "... A division ring need not have an identity by our definition (although some authors require it). An example is the set of complex numbers with ordinary addition but with multiplication z*w= z.conj(w) (the complex conjugate of w). ... A nonassociative division ring D need not have an identity element but is closely allied to one that does. The appropriate connection is "isotopy": every division algebra is isotopic to one with an identity. ... So just how many distinct division algebras are there of dimension n? ... the set of division algebras is an _open subet_ of R^(n^3). ... the isomorphism classes of division algebras ... corresponds to the quotient of this space by the action of GL(n,R) ... a Lie group of dimension n^2, leaving a set of dimension n^3-n^2 to parameterize the set of all isomorphism classes of division algebras. For n=1, ... R is the only division algebra ... For n=2, ... there is a 4-parameter family of division algebras ... For n=4: 48; for n=8: 448. ... On the other hand, many of these algebras are isotopic, that is, the new product may be found from the old one as x#y = P((Mx)(Ny)) where M, N, and P are fixed invertible matrices. Roughly speaking, this can provide at most a 3(n^2)-dimensional family of framed division algebras starting from any single one. ... [all of the] 4-parameter family of division algebras of dimension 2 ... are indeed ... isotopic to the complex field. ... [but] for n=4 or 8, this procedure will not exhaust the set of division algebras of dimension n, so that there exist algebras which are non-isotopic, and so more fundamentally different. ...". In 1962, Adams proved that the maximum number of linearly independent tangent vector fields on a sphere Sn is equal to n for (n+1) = 2, 4, or 8, and is less than n for all other n. (It had been proven earlier that the only parallelizable spheres are S1, S3, and S7.) As Okubo has noted in his book Introduction to Octonion and Other Non-Associative Algebras in Physics (Cambridge 1995), the theorem that real division algebras must have dimension 1,2,4,8 "...has been derived on the basis of topological reasoning on a seven-dimensional sphere. A pure algebraic proof of the theorem is still unknown." However, a closely related theorem is the N-square theorem: The product of two sums of N squares is a sum of N squares, over the real numbers, if and only if N = 1, 2, 4, or 8. There is a nice proof of the N-square theorem in the book Squares, by A. R. Rajwade (Cambridge 1993). Generalizing the N-square theorem, Rajwade defines a triple of integers (r,s,n) as ADMISSIBLE over a field K iff the product of a sum of r squares and a sum of s squares is a sum of n squares, over the field K. The N-square theorem is just that (N,N,N) is admissible over the real numbers for and only for: (1,1,1), corresponding to 1-dim real numbers; (2,2,2), corresponding to 2-dim complex numbers, which are not ordered; (4,4,4), corresponding to 4-dim quaternions, which are not commutative; and (8,8,8), corresponding to 8-dim octonions, which are not associative. (16,16,16) is NOT admissible, but (16,16,32) is admissible over the reals. Over the integral domain of the integers, 32 is the smallest n for which (16,16,n) is admissible, but over the real field it is only known (as of the time of Rajwade's book) that the smallest such n is in the range from 29 to 32.

In light of the theorems, I think that Octonions are uniquely fundamental and I have used them to build the D4-D5-E6-E7 model. However, many times I have been asked: IF 8-DIM OCTONIONS ARE SO GOOD, WOULD NOT 16-DIM THINGS BE EVEN BETTER? The 16-dim things that generalize octonions are called sedenions. The best reference for them known to me is NONASSOCIATIVE ALGEBRAS IN PHYSICS, by Jaak Lohmus, Eugene Paal, and Leo Sorgsepp Hadronic Press 1994, from which is taken the next figure, which uses notation {e0,e1,e2,e3,...,e15}. A concrete example of zero divisors in terms of that basis is given by Guillermo Moreno in q-alg/9710013: (e1 + e10)(e15 - e4) = -e14 - e5 + e5 + e14 = 0. (See the multiplication table of Lohmus et al, below). The basis used by Onar Aam is _ _ _ _ _ _ _ _ 1 i j k E I J K e i j k E I J K Sometimes I use a third notation, to avoid typing overbars: 1 i j k E I J K S T U V W X Y Z

The real numbers R correspond to the empty set, because R has no imaginary. In all cases, the empty set corresponds to 1. The complex numbers C correspond to a single point, a 0-dimensional simplex, because C has only one imaginary i . The quaternions correspond to a line segment, a 1-dimensional simplex, one end corresponding to i , the segment corresponding to j , and the other end corresponding to k . Going from i through j leads to k, and so forth cyclically, so that quaternion multiplication is determined by the associative triple cycle ijk which gives quaternion multiplication rules ii = jj = kk = ijk = -1 . Since there was no triple cycle for the complex numbers, the ijk associative triple cycle is new. It corresponds to the Lie algebra Spin(2) = U(1). The octonions correspond to a triangle, a 2-dimensional simplex, 3 vertices corresponding to I,J,K , 3 edges corresponding to i,j,k , and 1 face corresponding to E . There are 3+3+1 = 7 things. There are 7 (projective) lines each with 3 things. An octonion multiplication table, one of the 480 that exist, is then determined by the 7 associative triple cycles ijk -iJK -IjK -IJk EIi EJj EKk where -iJK means that -iJK = -1. For the octonions, 6 new associative triple cycles have appeared. They correspond to the Lie algebra Spin(4). The other 35 - 7 = 28 triples are not cycles. The sedenions correspond to a tetrahedron, a 3-dimensional simplex, 4 vertices v of the tetrahedron corresponding to EIJK ; 6 edges e of the tetrahedron corresponding to ijkTUV ; 4 faces f of the tetrahedron corresponding to WXYZ ; and the 1 entire tetrahedron T corresponding to S . There are 4+6+4+1 = 15 things. There are 35 (projective) lines each with 3 things. Geometrically, they are of the form: 4 like eee (where eee are all on the same face); 6 like vev (these are the edges); 12 like vfe (where v is opposite e on face f); 3 like eTe (where e is opposite e on the whole tetrahedron T); 4 like vTf (where v is opposite f on T); and 6 like fef (where the edge of e is not on f or f, that is, f and f are opposite to e). A sedenion multiplication table (there are even more than 480 of them) is then determined by the 35 associative triple cycles used by Onar Aam: ijk -iJK -IjK -IJk EIi EJj EKk __ __ __ __ __ __ __ -ijk iJK IjK IJk -EIi -EJj -EKk _ _ _ _ _ _ _ _ _ _ _ _ _ _ -ijk iJK IjK IJk -EIi -EJj -EKk __ __ __ __ __ __ __ -ijk iJK IjK IJk -EIi -EJj -EKk __ __ __ __ __ __ __ eEE eII eJJ eKK eii ejj ekk For sedenions, 28 new associative triple cycles have appeared. They correspond to the Lie algebra Spin(0,8). The other 455 - 35 = 420 triples are not cycles. Lohmus, Paal, and Sorgsepp give a table for the same sedenion multiplication product:

In the 16x16 table, the upper left 1x1 gives a table for R, the upper left 2x2 gives a table for C, the upper left 4x4 gives a table for Q, and the upper left 8x8 gives a table for O. Here is the same table in 1ijkEIJKSTUVWXYZ notation: 1 i j k E I J K S T U V W X Y Z 1 1 i j k E I J K S T U V W X Y Z i i -1 k -j I -E -K J T -S -V U -X W Z -Y j j -k -1 i J K -E -I U V -S -T -Y -Z W X k k j -i -1 K -J I -E V -U T -S -Z Y -X W E E -I -J -K -1 i j k W X Y Z -S -T -U -V I I E -K J -i -1 -k j X -W Z -Y T -S V -U J J K E -I -j k -1 -i Y -Z -W X U -V -S T K K -J I E -k -j i -1 Z Y -X -W V U -T -S S S -T -U -V -W -X -Y -Z -1 i j k E I J K T T S -V U -X W Z -Y -i -1 -k j -I E K -J U U V S -T -Y -Z W X -j k -1 -i -J -K E I V V -U T S -Z Y -X W -k -j i -1 -K J -I E W W X Y Z S -T -U -V -E I J K -1 -i -j -k X X -W Z -Y T S V -U -I -E K -J i -1 k -j Y Y -Z -W X U -V S T -J -K -E I j -k -1 i Z Z Y -X -W V U -T S -K J -I -E k j -i -1

Lohmus, Paal, and Sorgsepp note that if you use the Cayley-Dickson procedure to double the octonions to get the sedenions, you retain the properties common to all Cayley-Dickson algebras: centrality if xy = yx for all y in the algebra A, then x is in the base field of A, which is the real numbers R; simplicity no ideal K other than {0} and the algebra A, or, equivalently, if for all x in K and for all y in A xy and yx are in K, then K = {0} or A; flexibility (x,y,z) = (xy)z - x(yz) = -(z,y,x) or, equivalently, (xy)x = x(yx) = xyx ; power-associativity (xx)x = x(xx) and ((xx)x)x = (xx)(xx) or, equivalently, x^m x^n = x^(m+n) ; Jordan-admissibility xoy = (1/2)(xy + yx) makes a Jordan algebra; degree two xx - t(x)x + n(x) = 0 for some real numbers t(x) and n(x) ; derivation algebra G2 for octonions and beyond; and squares of basic units = -1 . For sedenions, you lose the following properties: the division algebra (over R) property xy = 0 only if x =/= 0 and y =/= 0 ; (A concrete example of zero divisors in terms of that basis is given by Guillermo Moreno in q-alg/9710013: (e1 + e10)(e15 - e4) = -e14 - e5 + e5 + e14 = 0.) linear alternativity (x,y,z) = (xy)z - x(yz) = (-1)P(Px,Py,Pz) where P is a permutation of sign (-1)P ; and the Moufang identities (xy)(zx) = x(yz)x (xyx)z = x(y(xz)) z(xyx) = ((zx)y)x . For sedenions, you retain the following properties: anticommutativity of basic units xy = -yx; and nonlinear alternativity of basic units (xx)y = x(xy) and (xy)y = x(yy).

The 28 new associative triple cycles of the sedenions are related to the 28-dimensional Lie algebra Spin(0,8), and to the 28 different differentiable structures on the 7-sphere S7 that are used to construct exotic structures on differentiable manifolds. Topological study of such manifolds has produced the only presently known proofs that the dimension of a division algebra must be 1, 2, 4, or 8. No algebraic proof is now known (Okubo (1995)). The only real Euclidean space with exotic differentiable structure is R4. In particular, R4 has many exotic differentiable structures # such that there exists a compact subset of any exotic R4# that cannot be smoothly embedded in ANY 3-sphere S3 in the R4#. In gr-qc/9405010 (also see gr-qc/9604048 by Carl Brans and hep-th/9604137 by J. Sladowski ), Carl Brans has suggested that, instead of looking at R4#, remove a point to get a semi-exotic cylinder R1 x# S3 where x# denotes topological (not differentiable) product. Brans remarks that an exotic spacetime R1 x# S3 could not have smooth differentiable structure forever, but that at some point on the time R1 axis, an obstruction would be encountered. Nobody knows what such an obstruction would look like. Another unknown is whether or not there exists an exotic S1 x# S3. Since the 4-dimensional spacetime of the D4-D5-E6-E7 physics model is of the form RP1 x S3, with topological structure S1 x S3, and exotic S1 x# S3 might be of physical interest. Before dimensional reduction, the D4-D5-E6-E7 physics model has 8-dimensional spacetime of the form RP1 x S7, with topological structure S1 x S7. Therefore, it is interesting to look atexotic structures on spheres, particularly S7. The Milnor spheres S(4k-1)#, of dimension 4k-1 for k=2 or greater, are homeomorphic to normal spheres S(4k-1) but not diffeomorphic to them. The 28 differentiable structures of the 7-sphere enabled John Milnor (Ann. Math. 64 (1956) 399) to construct exotic 7-spheres, denoted here by S7#. There are 27 exotic S7# spheres, plus one (the 28th) normal S7. A 7-sphere, whether exotic S7# or normal S7, can be "factored" by a Hopf fibration into a 3-sphere S3 and a 4-sphere S4. Each point of the S4 can be thought of as having one S3 attached. The Hopf fibration can be denoted by S3 - S7 - S4. Hopf fibrations can only be done for spheres of dimension 1,3,7,15: S0 - S1 - S1 (S0 = point) based on real numbers S1 - S3 - S2 based on complex numbers S3 - S7 - S4 based on quaternions S7 - S15 - S8 based on octonions Daniel Asimov has shown that the dimensions k = 0, 1, 3, and 7 are the only dimensions for which an open set in R^(2k+1) can be continuously filled by k-hoops. Consider S3 - S7 - S4 based on quaternions It is pretty clear that this is based on building the parallelizable S7, with coordinates ijkEIJK, from the associative ijk triple S3 and the coassociative EIJK square S4 of Onar Aam. Since the 3-sphere S3 is parallelizable, we can use ijk as a tangent coordinate system at any point and carry it around to any other point, so that ijk is a global coordinate system for the S3. Since the 4-sphere S4 is NOT parallelizable, if we put a tangent EIJK coordinate space at a particular point of the S4, we cannot carry it around to all other points of the S4. What we must do to define a "tangent" EIJK space at every point of S4 is to split S4 into two hemispheres, north and south, say, and put one EIJK tangent space on the northern hemisphere and another EIJK tangent space on the southern hemisphere. Notice that the two hemispheres meet at the equator. The equator of a 4-sphere is a 3-sphere. Since both northern and southern hemispheres have EIJK coordinate tangent spaces, you can choose IJK as the tangent coordinates for each of them, so that they agree on their common points, the S3 equator. This gives the normal 7-sphere S7, factored into ijk 3-sphere S3 and EIJK + EIJK 4-sphere S4 with IJK IJK equatorial S3 of S4. How do we get the exotic 7-spheres S7#? An S7# should be constructed by factoring into ijk 3-sphere S3 and STUV + WXYZ 4-sphere S4 with TUV XYZ equatorial S3 of S4. where STUV and WXYZ are not the same, but they must be mathematically consistent in that if S is a mirror reflecting ijk and TUV then W must also be a mirror reflecting ijk and XYZ, with the same orientation. How many different ways can this be done? It amounts to how many ways you can choose a triple out of the 8 STUV WXYZ 8! / 3! 5! = 8x7x6 / 3x2 = 56 and then choosing an orientation (half of the 56) to get 56 / 2 = 28 ways. IT IS ALSO EQUIVALENT TO ONAR AAM'S CONSTRUCTION OF 28 "NEW" triple cycles IN THE SEDENIONS. Carl Brans in gr-qc/9404003 conjectures (in the context of exotic R4#) that localized exoticness can act as a source of a physical field. In gr-qc/9906037, J. Sladowski uses isometry groups including SO(n,1) and SO(n,2) to show the validity of a form of the Brans conjecture: there are four-manifolds (spacetimes) on which differential structures can act as a source of gravitational force just as ordinary matter does. Since prior to dimensional reduction, the D4-D5-E6-E7 physics model has 8-dimensional spacetime of the form RP1 x S7 with topological structure S1 x S7, we can consider exotic structures on the spatial S7. As we just saw, there are 27 exotic S7# in addition to normal S7. If each Planck-size neighborhood of spacetime were a domain with its own S7-S7# structure, then each domain would act as a vertex in the 8-dim HyperDiamond lattice version of the D4-D5-E6-E7 model, and boundaries between domains would act as links. Since there are 28 different domain structures, isomorphic to 28-dim Spin(0,8), the boundaries/links would carry Spin(0,8) gauge bosons just as in the D4-D5-E6-E7 model prior to dimensional reduction. What about after dimensional reduction? The S1 x S7 spacetime structure is effectively reduced to S1 x S3 spacetime structure (with unknown exotic structures) plus a 4-dimensional coassociative internal symmetry space. If the internal symmetry space is regarded as S4, it has no exotic structure, and all the exociticity of S1 x S7 must go to the S1 x S3. If the internal symmetry space is regarded as S1x S3 or as R1 x S3 or as R4, then the internal symmetry space may have some exotic structure. This raises interesting questions whose answers may be useful in constructing physically realistic models.

SEDENIONS AND CLIFFORD ALGEBRAS: If they do not look at the whole sedenion algebra, but represent sedenions by their left or right adjoint actions, When Lohmus, Paal, and Sorgsepp get interesting matrix structures. To see how this works, first consider the octonion algebra: Let x and X be octonions, and let * denote octonion conjugation. Let Lx, *Lx, LX, and *LX be octonion left-actions. Let Rx, *Rx, RX, and *RX be octonion right-actions. As Dixon shows, the octonion left and right actions can be represented by 8x8 real matrices acting on the space of 1x8 real vectors, or the space of octonions. Consider the 7 matrices representing the imaginary octonions. The anticommutator of any two of them {Lp,Lq} = - 2 DELTA(pq) so that the 7 matrices generate the 128-dimensional Clifford algebra Cl(0,7), whose even subalgebra is 64-dimensional, whose minimal ideal spinor space OSPINOR is 8-dimensional. The 0-grade 1-dimensional scalar space of Cl(0,7) represents the octonion real axis. There is a 1 to 1 correspondence between the 1x8 minimal ideal OSPINOR on which OL acts by Clifford action, and the 1x8 octonion column vectors on which OL acts by matrix-vector action. This not only leads to triality in the larger Clifford algebra Cl(0,8) of Spin(0,8), but also to the division algebra property of octonions, because the map OL from OSPINOR to O is 1 to 1 and invertible. The space OR of octonion right-actions is equal to OL. Now - LOOK AT SEDENIONS: Lohmus, Paal, and Sorgsepp define sedenion left-actions SL by a 2x2 matrix of 8x8 matrices, which is the 16x16 matrix: OLx -*ORx *OLx ORx where *OL is the conjugate of OL and *x is the conjugate of x. They define sedenion right-actions SR by a 2x2 matrix of 8x8 matrices: ORx -OL*x OLx OR*x Thus, they represent the sedenion left and right actions SL and SR by 16x16 real matrices acting on 1x16 real vectors. Consider the 15 matrices representing the imaginary sedenions. The anticommutator of any two of them {Lp,Lq} = - 2 DELTA(pq) so that the 15 matrices generate the 32,768-dimensional Clifford algebra Cl(0,15), whose even subalgebra is 16,384-dimensional, whose minimal ideal spinor space SSPINOR is 128-dimensional. The 0-grade 1-dimensional scalar space of Cl(0,15) represents the sedenion real axis. There is an 8 to 1 correspondence between the 1x128 minimal ideal SSPINOR on which SL acts by Clifford action, and the 1x16 sedenion column vectors on which SL acts by matrix-vector action. This leads to failure of the division algebra property of sedenions, because the map SL from SSPINOR to S is 8 to 1 and invertible.

Consider SLx of sedenion left-multiplication by x as being represented by the 16x16 real matrix OLx -*ORx *OLx ORx and consider the 16x16 real matrices forming the 256-dim matrix algebra R(16), which is the Clifford algebra Cl(0,8) of Spin(0,8): 0 2 2 2 2 2 2 2 7 5 5 5 5 5 5 5 4 4 2 2 2 2 2 2 5 7 5 5 5 5 5 5 4 4 4 2 2 2 2 2 5 5 7 5 5 5 5 5 4 4 4 4 2 2 2 2 5 5 5 7 5 5 5 5 4 4 4 4 4 2 2 2 5 5 5 5 7 5 5 5 4 4 4 4 4 4 2 2 5 5 5 5 5 7 5 5 4 4 4 4 4 4 4 2 5 5 5 5 5 5 7 5 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 7 1 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 3 1 3 3 3 3 3 3 6 4 4 4 4 4 4 4 3 3 1 3 3 3 3 3 6 6 4 4 4 4 4 4 3 3 3 1 3 3 3 3 6 6 6 4 4 4 4 4 3 3 3 3 1 3 3 3 6 6 6 6 4 4 4 4 3 3 3 3 3 1 3 3 6 6 6 6 6 4 4 4 3 3 3 3 3 3 1 3 6 6 6 6 6 6 4 4 3 3 3 3 3 3 3 1 6 6 6 6 6 6 6 8 The numbers refer to the grade in Cl(0,8) of the matrix entry: grade 0 1 2 3 4 5 6 7 8 dimension 1 8 28 56 70 56 28 8 1Important Notation Notes: The two blocks of the form 0 2 2 2 2 2 2 2 4 4 2 2 2 2 2 2 4 4 4 2 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 2 2 2 4 4 4 4 4 4 2 2 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 4 are more symbolic than literal. They mean that: the 28 entries labelled 2 correspond to the antisymmetric part of an 8x8 matrix; the 35 entries labelled 4 correspond to the traceless symmetric part of an 8x8 matrix; and the 1 entry labelled 0 corresponds to the trace of an 8x8 matrix. A more literal, but more complicated, representation of the graded structure of those two blocks is: 0 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 However, in the more literal representation, the entries are not all independent. The more symbolic representation is a more accurate reflection of the number of independent entries of each grade. The two blocks of the form 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 can be taken more literally, as they mean that: the 8 entries labelled 1 correspond to the diagonal part of an 8x8 matrix; and the 56 entries labelled 3 correspond to the off-diagonal part of an 8x8 matrix.The conventions of the above Notation Notes are used from time to time in my papers and web pages.The even subalgebra Cle(0,8) of Cl(0,8) is then the block diagonal 0 2 2 2 2 2 2 2 4 4 2 2 2 2 2 2 4 4 4 2 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 2 2 2 4 4 4 4 4 4 2 2 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 4 4 4 4 4 4 4 6 6 4 4 4 4 4 4 6 6 6 4 4 4 4 4 6 6 6 6 4 4 4 4 6 6 6 6 6 4 4 4 6 6 6 6 6 6 4 4 6 6 6 6 6 6 6 8 The SLx matrix action of sedenion left-multiplication by x restricted to the block diagonal of the even subalgebra Cle(0,8) is then OLx ORx and the block diagonal part of the SL matrices is just the direct sum OL + OR each of which is an 8x8 real matrix acts on 8-dimensional vector space isomorphically to its action of 8-dimensional spinor space OSPINOR. Denote the OL spinor space by OSPINOR+ and the OR spinor space by OSPINOR-. Then, the direct sum OSPINOR+ + OSPINOR- represent the +half-spinor space and the -half-spinor space of the Clifford algebra Cl(0,8) of Spin(0,8) + + + + + + + + - - - - - - - - The +half-spinor space OSPINOR+ is acted on by the OL elements of Cle(0,8) of grade 0 2 4 dimension 1 28 35 while the -half-spinor space OSPINOR- is acted on by the OR elements of Cle(0,8) of grade 4 6 8 dimension 35 28 1 we have the useful result that the block diagonal part of the adjoint left action SL of sedenions represents the 16-dimensional full spinor representation of the Clifford algebra Cl(0,8) of the Lie algebra Spin(0,8).

ONAR AAM has shown more details of the relationship between SEDENIONS and CLIFFORD ALGEBRAS. Consider the set of N points, or nodes. Now consider the set of all subsets of the N nodes. It has 2^N elements. Now, from the Yang Hui (Pascal) triangle, we know that: 2^N = 1 + N + N(N-1)/2 + ... + N! / (k! (N-k)! + ... + N + 1 Now ask how many of the subsets have k nodes. It is the term N! / (k! (N-k)! In particular, the number of edges of a fully connected graph of the N nodes is the number of 2-node subsets, because each edge just connects 2 nodes. What has this to do with Clifford algebras? A Clifford algebra of an N-dim vector space looks a lot like the "set of subsets of N things". The Clifford algebra is called a "graded" algebra because it elements are not all on an equal footing. Its elements belong to various levels, or "grades". Its elements that correspond to the subsets containing k things are called the k-grade elements of the Clifford algebra. Geometrically, the k-grade elements of the Clifford algebra Cl(0,N) of the N-dimensional vector space are identified with the k-dimensional subspaces of the N-dimensional vector space. That is why Clifford, when he invented Clifford algebras, called them "geometric algebras". They encode all the relationships among the subspaces of the N-dimensional vector space. ----------- Here are 2 technical qualifications: We are dealing with real vector spaces, and so with real Clifford algebras, just as we have been dealing with division algebras over the real numbers; and we are ignoring signature of the vector space. Signature can be put in without problem, except it makes the discussion more complicated, so we don't do it here. ----------- Particularly, the 0-grade Clifford algebra element 2^N = 1 + N + N(N-1)/2 + ... + N! / (k! (N-k)! + ... + N + 1 is just 1, the origin of the vector space. There are N of the 1-grade elements, and they are a basis of the N-dimensional vector space. There is 1 N-grade element, and it is an N-dimensional volume element. There are N(N-1)/2 of the 2-grade elements. They are pairs of vectors, and are called bivectors. In the Clifford algebra, the product of two vectors is of grade 2, and is a 2-vector, also it is a called a bivector. The Clifford product of a 1-vector and a k-vector (elements of grade 1 and of grade k) is the difference between a (k+1)-vector and a (k-1)-vector. The k+1 part of the Clifford product looks like the exterior product, or generalized cross-product, and the k-1 part looks like an interior or dot product. In this way, the Clifford product combines both of the useful geometric vector products. What about the product of two bivectors, each of grade 2? The Clifford product is calculated in two stages. Call the bivectors ab and xy (where a,b,x,y are 1-vectors) The first stage of the calculation is to use ab xy = a (b xy) and calculate b xy Since b is a 1-vector and xy is a bivector, the product b xy lives in 2 of the grades: part of it is in grade 2+1 = 3 part of it is in grade 2-1 = 1 Denote these parts by bxy3 and bxy1 The second stage of the calculation is to get a (bxy1 + bxy3) part of it is in grade 1-1 = 0 part of it is in grade 1+1 = 2 part of it is in grade 3-1 = 2 part of it is in grade 3+1 = 4 Now, it looks like we have a mess. The Clifford product AB of two bivectors A and B is partly the sum of two bivectors, and so is partly in the bivector 2-grade subspace, but it also has a 0-vector part and a 4-vector part. HOWEVER, we can look at the COMMUTATOR of the Clifford product of two bivectors. The commutator is [A,B] = (1/2)(AB - BA) When we look at this, we see that the 0-vector parts and the 4-vector parts CANCEL OUT, and that we have: THE COMMUTATOR [A,B] OF TWO BIVECTORS A and B IS ALSO A BIVECTOR. That is, the Clifford algebra bivectors form a SUBALGEBRA of the Clifford algebra, under the commutator product. This subalgebra is the LIE ALGEBRA called Spin(0,N), and it is the simply connected covering algebra of the algebra that generates the rotation group of the N-dimensional vector space of Cl(0,N). What about other subalgebras of Cl(0,N)? One thing you will see is that the set of all even-number grades closes under the Clifford product. This is called the EVEN SUBALGEBRA of Cl(0,N), and is denoted Cle(0,N). Another thing is that the Clifford algebra structure 2^N = 1 + N + N(N-1)/2 + ... + N! / (k! (N-k)! + ... + N + 1 is symmetric in that the elements of grade k are in 1-1 correspondence with elements of grade N-k This is called HODGE DUALITY. Hodge duality lets you define vector cross-products in certain dimensions (1,3,7). It is clear that the 1-vectors by themselves do not form an algebra directly, because the Clifford product of two vectors is partly in grade 1+1 = 2 and partly in grade 1-1 = 0 However, you can just look at the part that is in grade 1+1 = 2. For the 3-dimensional vector space of Cl(0,3) 2^3 = 1 + 3 + 3 + 1 the 1-vectors correspond by Hodge duality to the 2-vectors, so you can use the duality to define the 3-dimensional cross-product. For the 7-dimensional vector space of Cl(0,7) 2^7 = 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 you need to take an extra step. You must pick out an "associative 4-grade element" (call it A4 - it is related to a quaternionic subspace). Then, as for 3-dim, look only at the 2-grade bivector part of the product of two 1-vectors. Call it xy2. Then, take the product of xy2 and A4. It has a grade 4-2 = 2 part and a grade 4+2 = 6 part. Look only at the 6-grade part. By Hodge duality, the 6-grade part corresponds to a unique 1-vector of grade 7-6 = 1. That is the 7-dimensional cross product. It only works because the coassociative quaternionic subspace of the 7-dim vector space is in one of the "middle" grades (35-dimensional things). Note that there are two 35-dimensional things: one of grade 3 (like Onar Aam's associative triple cycles); and one of grade 4 (like Onar Aam's coassociative squares). THIS IS ONAR AAM'S NEW GEOMETRIC INTERPRETATION of what are called ASSOCIATIVE and COASSOCIATIVE CALIBRATIONS. They are NOT widely known, and are written up only in a few books and papers. One such book is Spinors and Calibrations, by F. Reese Harvey (Academic Press 1990). (See particularly around pp. 144-145. It is a good book.) Calibrations are used in dimensional reduction of spacetime in the D4-D4-E6 physics model. Consider Cl(0,8) 2^8 = 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 and its even subalgebra Cle(0,8): 1 + 28 + 70 + 28 + 1 We have seen that the 28-dim bivector part form the 28-dimensional Lie algebra Spin(0,8) of antisymmetric 8x8 real matrices under [A,B] = (1/2)(AB - BA) What about the symmetric 8x8 real matrices under AoB = (1/2)(AB+BA)? They form the 64-28 = 36-dimensional Jordan algebra J8(R). Can 36-dim J8(R) be represented by half of 1 + 70 + + 1 similarly to the way Spin(0,8) is represented by half of 28 + 28 To explore this question, note that Hodge duality splits the 70-dim 4-vector space into two 35-dim spaces: 1 + 35+35 + + 1 Also note that 4-forms dual to the 4-vectors give coassociative Cayley calibrations PHI and that 8-forms give volume elements VOL, and that for s and t in the 16-dim space S of Cl(0,8) spinors, the tensor product s x t decomposes into antisymmetric s /\ t in Cl(0,8) subspaces of grades 1,2,5,6 and symmetric s o t in Cl(0,8) subspaces of grades 0,3,4,7,8 Further, the square of a spinor s in S is symmetric s x s in Cl(0,8) subspace of grades 0,4,8: s x s = s o s = 1 + PHI + VOL Reese Harvey (Spinors and Calibrations (Academic Press 1990)) shows (in exercise 6-17) that the 14-dim exceptional Lie Group G2, the automorphism group of the octonions O, can be represented in the graded exterior algebra of forms on octonion imaginary space ImO = S7 /\(ImO)* whose structure is dual to that of Cl(7) 2^7 = 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 In particular, the representation space /\^2(ImO)* decomposes as the direct sum /\^2_{7} + /\^2_{14} where /\^2_{7} is Im(O) and /\^2_{14} is the Lie algebra G2 characterized by the kernel of the derivation map from 49-dim GL(ImO) to 35-dim /\^3(ImO)* (It is the antisymmetric part of GL(ImO), less S7>) and /\^3(ImO)* decomposes as /\^2_{1} + /\^2_{7} + /\^2_{27} where /\^2_{7} is Im(O) and /\^2_{1} + /\^2_{27} is the symmetric part of GL(ImO) (It is the 1-dim scalar plus the 27-dim Jordan algebra J3(O).) ----------------------------------------------- Now (at last) we can see what all this has to do with sedenions. The sedenion multiplication table is 16x16 so it has 256 = 2^8 entries and can be written as a 16x16 matrix: r i j k E I J K S T U V W X Y Z r x x x x x x x x x x x x x x x x -i x x -j x x q -k x x -E x x o o o -I x x o o -J x x o -K x x -S x x s s s s s s s -T x x s s s s s s -U x x s s s s s -V x x s s s s -W x x s s s -X x x s s -Y x x s -Z x x The 16+15+15 = 46 x entries denote the "real" products that cannot belong to an associative triple cycle of the type ijk. For the ri part of the table, the complex numbers, there are no associative triple cycles. For the rijk part of the table, the quaternions, there is only one associative triple cycle, the ijk triple itself, denoted by the q entry. It occupies the upper triangular part of the lower right quadrant, and so is only 1/2 (from picking the upper triangular part) of 1/3 (from picking the lower right quadrant out of the 3 new quadrants that come from extending the complex numbers to the quaternions). That is equivalent to considering the overlapping arising from the 6 permutations of the one associative triple cycle ijk, for which the first two elements would be ij,jk,ki,ji,kj,ik. The 6 o entries represent the 6 new associative triple cycles that come with the octonions. The 28 s entries represent the 28 new associative triple cycles that come with the sedenions. Recall that the block diagonal part of SL sedenion left-multiplication corresponds to the full spinor representation of the 16x16=256-dim Clifford algebra Cl(0,8): 2^8 = 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 The 28-dimensional bivector space represents the 28-dimensional Lie algebra Spin(0,8), using the commutator product. It is called the 28-dimensional adjoint representation of Spin(0,8). The 8-dimensional vector space represents the vector space on which the Lie group Spin(0,8) acts as a rotation group. It is called the 8-dimensional vector representation of Spin(0,8). Each bivector, or Lie algebra element, can be represented by a "new" sedenion associative triple. THIS IS A NEW WAY TO LOOK AT LIE ALGEBRAS. Now look at the even subalgebra Cle(0,8) of Cl(0,8). Since Cl(0,8) is given by 2^8 = 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 Cle(0,8) is given by 1 + 28 + 70 + 28 + 1 Now pick a 7-dimensional subspace (imaginary octonions) of the 8-dimensional vector space (octonions), and look at the rotations (Spin(0,8) bivectors) whose axes are those 7 vectors. That lets us pick out 7 of the 28 Spin(0,8) bivectors and regard them as vectors in a 7-dimensional space. The other 21 bivectors then represent rotations in that 7-dimensional space, that is, they represent elements of Spin(0,7), the Lie algebra of the Cl(0,7) Clifford algebra. Therefore, Cle(0,8) then can be written as: 1 + 7 + 21 + 70 + 21 + 7 + 1 We have already seen that the Spin(0,7) Clifford algebra can be written as: 2^7 = 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 So we can see, by splitting the 70-dimensional 4-vector space of Cl(0,8) into two spaces, one 35-dimensional associative 3-vector space of Cl(0,7), and one 35-dimensional coassociative 4-vector space of Cl(0,7), that Cle(0,8), the even subalgebra of Cl(0,8), is just Cl(0,7), the real 7-dimensional Clifford algebra. Going to lower and lower dimensions, we can see that the 6 o entries representing the 6 new associative triple cycles that come with the octonions represent the 6-dimensional Lie algebra Spin(0,4), or rotations in 4-dimensional spacetime, which looks like 3-dimensional spatial rotations plus 3-dimensional Lorentz boosts. Also, the one q entry representing the associative quaternion triple represents the 1-dimensional Lie algebra Spin(0,2) = U(1), the rotations in the complex plane. WHAT ABOUT GOING UP TO HIGHER DIMENSIONS? For 32-ons, we get 120 new associative triple cycles, and they represent the Lie algebra Spin(0,16) of the Clifford algebra Cl(0,16). HOWEVER, NOTHING REALLY NEW HAPPENS BECAUSE OF THE PERIODICITY PROPERTY OF REAL CLIFFORD ALGEBRAS. The periodicity theorem says that Cl(0,N+8) = Cl(0,8) x Cl(0,N) (here x = tensor product) That means that the Clifford algebra Cl(0,16) of the 32-ons is just the tensor product of two copies of the Clifford algebra Cl(0,8). So, everything that happens in the 32-on Clifford algebra is just a product of what happens with Cl(0,8). That point is emphasized by the fact (see Lohmus, Paal, and Sorgsepp, Nonassociative Algebras in Physics (Hadronic Press 1994)) that the derivation algebra of ALL Cayley-Dickson algebras at the level of octonions or larger, that is, of dimension 2^N where N = 3 or greater, is the exceptional Lie algebra G2, the Lie algebra of the automorphism group of the octonions. The exceptional Lie algebra G2 is 14-dimensional, larger than the 8-dimensional octonions, but smaller than the 16-dimensional sedenions. Each 8-dimensional half-spinor space of Cl(0,8) has a 7-dimensional subspace of "pure" spinors (see Penrose and Rindler, Spinors and space-time, vol. 2, Cambridge 1986) that correspond by triality to the 7-dimensional null light-cone of the 8-dim vector space. The 7-dimensional spaces are 7-dimensional representations of the 14-dimensional Lie algebra G2.

With respect to the D4-D5-E6-E7 physics model, the block diagonal part of the SL sedenion left-multiplication matrix represents the 16-dimensional full spinors of Cl(0,8), whose bivector Lie algebra is 28-dim D4 Spin(0,8). You can then make the conformal group, 45-dim D5 Spin(1,9), extending 28-dim D4 Spin(0,8), by forming the 2x2 matrices over Cl(0,8), which are the Clifford algeba Cl(1,9), as described by Ian Porteous in chapters 18 and 23 of Clifford Algebras and the Classical Groups (Cambridge 1995). Note that there is a Spin(0,2) = U(1) symmetry due to the 2x2 matrices that can be used to give the vector extension to D5 from D4 a complex structure. The conformal group, 78-dim E6, extending 45-dim D5 Spin(1,9), may be formed in an exceptional way, by constructing E6 commutation relations WITHIN the Cl(1,9) Clifford algebra, by going from the Spin(1,9) group of Cle(1,9) to the Pin(1,9) group of Cl(1,9). The effect is to add to 45-dim Spin(1,9) the 32-dim full spinors of Cle(1,9) plus a Spin(0,2) = U(1) symmetry arising from the 2-fold covering of Cle(1,9) by Cl(1,9). The Spin(0,2) = U(1) symmetry gives the spinors a complex structure. (Such a construction cannot be done in all dimensions. It is exceptional, and is due to underlying octonion structures.) Then, as discussed by Gilbert and Murray in Clifford algebras and Dirac operators in harmonic analysis (Cambridge 1991), Dirac operators and related structures can be constructed and used to build the D4-D4-E6 physics model.

Finally, to answer the initial question: WHY NOT SEDENIONS? to me, the useful part of the sedenion product is not the full sedenion product, but only the block diagonal part OLx ORx of the SL sedenion left-multiplication matrix, because it represents the 16-dimensional full spinors of Cl(0,8) and it is equivalent to the direct sum OL + OR of octonion left-multiplication and octonion right-multiplication. Sedenions may also be a good way to express the structure of the last Hopf fibration S7 --> S15 --> S8. Since there are no Hopf fibrations other than S0 --> S1 --> RP1 = Spin(2)/Spin(1) S1 --> S3 --> S2 = CP1 = SU(2)/U(1) = Sp(1)/U(1) = Spin(3)/Spin(2) S3 --> S7 --> S4 = HP1 = Sp(2)/(Sp(1)xSp(1)) = Spin(5)/Spin(4) S7 --> S15 --> S8 = OP1 = Spin(9)/Spin(8) sedenion structure may show why the sequence terminates. Also, it is possible that the sedenions may show the structure of the 16-dimensional octonionic projective plane OP2 = F4/Spin(9) and may also show why there is no higher-dimensional octonionic projective space. I think that the highest and best use of sedenions is not for the full algebra structure but as a very useful expression of "global" organization of the structure of octonions. Other useful 16-dimensional structures are the 16-dim Barnes-Wall lattice /\16 and the XY-product of Geoffrey Dixon.

Another way to see why sedenions do not make Division Algebras is to look at the construction of Division Algebras from Clifford Algebras.

Sedenions and other ZeroDivisor Algebras have been discussed by Charles Muses.

Mari Imaeda has an interesting Octonion and Sedenion web site.

Tony Smith's Home Page ......