Helmut Eschrig Leibniz-Institut für Festkörper- und Werkstofforschung Dresden Leibniz-Institute for Solid State and Materials Research Dresden Hopf Fibrations Consider a classical magnetization field in R 3 which is longitidinally stiff and transversally soft. It is given by M : R 3 S : x M(x), M = const. i M 0 causes some positive energy density, so an excitation with finite energy must have i M localized, M const. for r, which means that M is unique on the compactified space R 3. With the stereographic projection, R 3 S 3, one may consider M : S 3 S. Are there topologically stable such mappings?
Introduce polar coordinates (r, θ, φ,...,φ n ) in R n+1 S n with r: radius, θ: polar angle, φ i : azimuthal angles. Stereographic projection S n R n : y θ θ/ cot(θ/) x x = cot θ, φ i = φ i, i =,...,n. The projection is a conformal bijection, it preserves angles. It maps S n k S n into S n k R n, if S n k does not contain the pole of projection of S n. It maps S n k S n into R n k R n, if S n k does contain the pole of projection of S n (infinite point of R n ). Recall that a (linear) Algebra A over a field F is a vector space A over the field F of scalars, endowed with a bilinear multiplication; for all x, y, z A and a, b F it holds that x(ay + bz) = axy + bxz, (ay + bz)x = ayx + bzx. A is unital, if it contains a unity e with xe = ex = x, x A. Embedding F A : F a ae A. A is associative, if (xy)z = x(yz), x, y, z A. A is commutative, if xy = yx, x, y A. A is a division algebra, if xy = 0 only, if x = 0 or y = 0. A is normed, if there is a vector space norm : A R + with xy = x y (more generally xy C x y ).
Let A be a unital algebra over the field F, endowed with an involution : A A : x x, Let Define x = x, (x + y) = x + y, (ax) = a x, (xy) = y x. x + x = tr(x) F, xx = n(x) F. à = A A (x 1, x ), (x 1, x )(y 1, y ) = (x 1 y 1 y x, x 1 y + y 1 x ), (x 1, x ) = (x 1, x ). A à : x (x, 0), tr(x 1, x ) = (x 1, x ) + (x 1, x ) = (x 1 + x 1, 0) = (tr(x 1 ), 0) ˆ= tr(x 1 ), n(x 1, x ) = (x 1, x )(x 1, x ) = (x 1 x 1 + x x, x 1 x + x 1 x ) = (n(x 1 ) + n(x ), 0) ˆ= n(x 1 ) + n(x ). Base (e, {e n }) of vector space Ã: dim à 1 à x = ae + a n e n, ex = xe = e, n=1 x + x = a(e + e ) + dim à 1 n=1 e = e, e n = e n. a n (e n + e n) F, Cayley-Dickson process A à Ã. A = R x = ae, a R, e = 1, is a unital, associative, commutative, normed division algebra, dim A = 1. à z = (x 1, x ) = (x 1, 0) + (0, x ) x 1 = ae, x = a 1 e 1. e 1 = i, tr(z) = x 1, n(z) = x 1 + x. à = C is a unital, associative, commutative, normed division algebra, dim A =. ( ) ( ) z z z = z = 1. n(z) n(z) A = C z = ae + a 1 e 1 = a + a 1 i. à h = (z 1, z ) = (z 1, 0) + (0, z ) ˆ= (x 1, x, 0, 0) + (0, 0, x 3, x 4 ), (z 1, z )(z 1, z ) = (z 1, z )(z 1, z ) = (z 1 z 1 + z z, 0) = (x1 + x + x 3 + x 4, 0, 0, 0), e = e i e i = e i = e, e i e j + e j e i = (e i e j + e j e i ) = 0. e 1 e = (i, 0)(0, 1) = (0, i) = e 3, e i e j = ε k ij e k, ε k ij = δ ijk 31. Irreducible representation: e = 1, e i = iσ i : Pauli matrices à = H: quaternion algebra is a unital, associative, non-commutative, normed divison algebra, dim A = 4 (see supplementary material at the end). ( h h n(h) ) = ( h n(h) ) h = 1.
A = H h = ae + a 1 e 1 + a e + a 3 e 3. Ã o = (h 1, h ) = (h 1, 0) + (0, h ) ˆ= (x 1,...,x 8 ). (h 1, h )(h 1, h ) = (x1 + + x 8, 0, 0, 0, 0, 0, 0, 0). e = e i e i = e i = e, e i e j + e j e i = (e i e j + e j e i ) = 0. For instance (e 1 e )e 4 = e 3 e 4 = (ê 3, 0)(0, 1) = (0, ê 3 ) = e 8, e 1 (e e 4 ) = e 1 e 7 = (ê 1, 0)(0, ê ) = (0, ê 1 ê ) = (0, ê 3 ) = e 8. Ã = O: octonion algebra (Cayley numbers) is a unital, non-associative, non-commutative, normed divison algebra, dim A = 8. ( o ) ( o o = n(o) n(o) ) o = 1. o 1 o = o 3, { o 0 o 1 = o 3 o /n(o ), o 1 0 o = o 1 o 3/n(o 1 ). The next step yields the algebra S of sedenions. Now, there are zero divisors in S, which latter is a unital, non-associative, non-commutative, normed algebra, but no division algebra any more and therefore of much less interest. There is some relation to the exceptional Lie group G, the automorphism group of the octonion algebra. Hurwitz s theorem: Every normed division algebra over the real numbers is isomorphic to R, C, H, or O. Frobenius theorem: Every finite-dimensional associative division algebra over the real numbers is isomorphic to R, C, or H. (As a ring, such an algebra is a skew-field.) Up to isomorphisms the only finite-dimensional associative-commutative division algebras over the real numbers are R and C. (As a ring such an algebra is a field.)
Hopf mappings are homotopically non-trivial continuous mappings of spheres into spheres of lower dimension where the preimage of every image point is another sphere embedded in the domain sphere of the mapping. There are only four cases: S 0 S 1 S 1, S 1 S 3 S, S 3 S 7 S 4, S 7 S 15 S 8, The unit ball B n in the real vector space R n is B n = { x } x 1, n = 1,,.... The unit sphere S n 1 is the boundary of the unit ball B n, S n 1 = { x x = 1 }. The trivial case is B 1 = [ 1, 1], S 0 = { 1, 1}. Heinz Hopf, 1931, found the second one: H. Hopf, Über Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, Math. Ann. 104, 637 665 (1931). B is the disc, S 1 the circle, B 3 the ball, S the sphere,... S 1 S 3 π S, The trivial case, S 0 S 1 π S 1, is just given by π(z) = z, z = 1, z C. It has degree of mapping (winding number). p : H C R 4 R 3 : p(h) = p(z 1, z ) = ( z 1 z, z1 z + z 1 z, i(z 1 z z 1 z )). { } { } S 3 = h H n(h) = 1 = (z 1, z ) C z1 + z = 1. Parametrize on S 3 z 1 = cos θ ( exp i ψ + φ ), z = sin θ ( exp i ψ φ ), then p(z 1, z ) = ( cos θ, sin θ cos φ, sin θ sin φ ), hence, π = p S 3. Obviously, π(z 1, z ) = π(e iζ z 1, e iζ z ), so that π 1 (h) S 1.
It is not difficult to show S 3 (z 1, z ) z = z 1 z 1 C R stereographic projection S. Stereographic projection: y θ θ/ cot(θ/) z = z 1 /z = cot θ, Arg z = Arg z 1 Arg z = φ. The only invariant of (z 1, z ) z 1 /z is z 1 e iζ z 1, z e iζ z, and the stereographic projection is one-one, hence π 1 (h) = S 1. z The unit sphere S is clearly not contractible (continuously in itself into a point). Since the Hopf mapping S 3 S is continuous and onto, it is homotopically non-trivial (not continuously deformable into a constant mapping to a single point). The corresponding homotopy group is hence also non-trivial. π 3 (S ) = Z. Historically, the Hopf mapping was the first proof of a non-trivial homotopy group for mapping a sphere into a sphere of lower dimension. (The computation of homotopy groups π k (S n ) for k > n is notoriously difficult and far from being completed.) Consider Hopf s mapping in some more detail. Take also a stereographic mapping S 3 R 3. For more clarity use coordinates (u, z) = (u, v 1, v ) in S and (z 1, z ) = (y, x 1, x, x 3 ) in S 3. p 1 (1, 0, 0) = { (z 1, z ) z 1 = 1, z = 0 }. This is the meridian of S 3 in the (y, x 1 )-plane of R 4 which sterographically maps to the x 1 -axis of R 3. p 1 ( 1, 0, 0) = { (z 1, z ) z = 1, z 1 = 0, hence y = 0 }. This is an equator of S 3 which projects to the unit circle around the origin in the (x, x 3 )-plane of R 3. The two circles in R 3 are obviously linked: x 3 p 1 (1, 0, 0) x x 1 p 1 ( 1, 0, 0) Since the stereographic projection is a homeomorphism, the corresponding two circles on S 3 are also linked. (On S 3 a meridian and an equator need not have common points; here, on the former z 1 0, z = 0 while on the latter z 0, z 1 = 0. The equatorial hyperplane is 3-dimensional.)
For any other point (u, v 1, v ) S with u 1 obviously z 1 0 z, hence p 1 (u, v 1, v ) does not intersect the above two circles of R 3. Now, p 1 (u, v 1, v ) = (e iζ z 10, e iζ z 0 ) where (z 10, z 0 ) is any chosen point on p 1 (u, v 1, v ). There is ζ 0 and ζ 0 + π for which z 1 is real and hence the points have y = ±y 0 0, x 1 = 0. They are stereographically projected to the (x, x 3 )-plane, that with y > 0 outside of the circle p 1 ( 1, 0, 0) and that with y < 0 inside. Hence, p 1 (u, v 1, v ) and p 1 ( 1, 0, 0) are again linked. Let A and B be two arbitrary distinct points on S. Rotate the coordinates of S 3 so that p 1 (A) is in the (x, x 3 )-plane; the circle p 1 (B) is linked with that circle. S 3 S 7 π S 4, S 7 (h 1, h ) h = h 1 h 1 H R4 stereographic projection S 4. S 7 S 15 π S 8, S 15 (o 1, o ) o = o 1 o 1 O R8 stereographic projection S 8. For the Hopf mapping, any two preimages of distinct points on S form pairs of linked circles on S 3. This process cannot be continued since there is no mapping with sedenions, (s 1, s ) s = s 1 s 1, any more. The algebra S of sedenions is no division algebra and hence s 1 is not unambiguously defined any more.
A (continuous) principal fiber bundle G P π M consists of a manifold P, the total space, a manifold M, the base space, and a Lie group G, the typical fiber. P is locally trivial; for every point x M there is a neighborhood U, x U M, so that P U = U G is a topological product space and π : P U U is the corresponding projection. For every x M, the fiber over x is π 1 (x) G. G P π M is also called a fibration of the manifold P; P is obtained by clueing of trivial patches P U along fibers. In the degenerate case of the real Hopf fibration S 0 S 1 S 1 the two point set S 0 = { 1, 1} with the discrete topology may be considered as a Lie group O(1) = Z = { 1, 1} with group operations 1 1 = 1 1 = 1, 1 1 = 1 1 = 1. The connected Lie group of norm conserving automorphisms of R n is SO(n). Its isotropy subgroup which keeps a given point x R n fixed is isomporphic to SO(n 1) which is a (non-normal) subgroup of SO(n), hence the orbit space SO(n)/SO(n 1) is a homogeneous space diffeomorphic to the sphere S n 1, n > 1, S n 1 SO(n) SO(n 1). An analogous consideration in C n yields and in H n, S n 1 S 4n 1 SU(n) SU(n 1), Sp(n) Sp(n 1). Sp(n) = Sp(n, C) U(n) is the compact symplectic group. SO(1) = SU(1) = Sp(1) = {1}, also SO(4) SO(3) SO(3). SO(n) = SO(n, R), SU(n) = SO(n, C), Sp(n) = SO(n, H). The latter relation led Hamilton to the concept of quaternions. Connections on principal fiber bundles have a one-one translation to local gauge field theories (e.g. with skyrmions). U(1) S 3 S contains the Dirac monopole. By considering tori T in S 3 it is also relevant for ferroelectrica and in the QHE. SU() S 7 S 4 contains the Belavin-Polyakov instanton. Again with tori it has relevance for 3D topological insulators. It has also relevance for pairs of q-bits. Adams-Atiyah theorem: The four considered cases are the only principal fiber bundles where total space, base space and fiber are all spheres.
Supplementary Material If generally tr(xy) = tr(yx) in A, then the same holds in Ã. Proof: x, y Ã, x i, y i A. tr(xy) = (x 1, x )(y 1, y ) + (y 1, y )(x 1, x ) = = (x 1 y 1 y x, x 1 y + y 1 x )+ + (y 1 x 1 x y, y 1x x 1 y ) = = tr(x 1 y 1 ) tr(y x ) = tr(y 1x 1 ) tr(x y ) = = tr(yx). tr(xy) = tr(yx) in every Cayley-Dickson process which starts with an algebra A for which the trace is cyclic. If tr(xy) = tr(yx), then xy = x y. Proof: xy = (x 1 y 1 y x, x 1 y + y 1 x ) = = (x 1 y 1 y x )(y 1 x 1 x y )+ + (x1 y + y 1 x )(y x 1 + x y 1 ) = = x 1 y 1 x 1 y 1 x y y x y 1 x 1 + x y + + x 1 y + x1 y x y 1 + y 1x y x 1 + x y 1 = = ( x 1 + x )( y 1 + y ) tr(x 1 y 1 x y ) + tr(y 1x y x 1) = = x y.