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On Various Geometries Giving a Unified Electric and by Thomas J.M.

By Thomas J.M.

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S 2 //. It thus defines a complex vector bundle on S 2 . x/ D 1 for all x 2 S , we have in fact a complex line bundle over S 2 . It can be shown that it is the line bundle associated to the Hopf fibration 2 S 1 ! S 3 ! S 2: Incidentally, e induces a map f W S 2 ! x/, which is one-to-one and onto. C/. This example can be generalized to higher-dimensional spheres. C/ satisfying the Clifford algebra relations [103] i j C j i D 2ıij for all i; j D 1; : : : ; 2n C 1. x/ D 1 for all x 2 S 2n , so that e D vector bundle over S 2n .

I ˝ S/ D Á" W H ! 22) where I denotes the identity map. G/ is a commutative Hopf algebra. h/ D h ˝ h; h ¤ 0: The general definition of a Hopf algebra is as follows. Let H be a unital algebra and let m W H ˝ H ! H and Á W C ! H denote its multiplication and unit maps, respectively. 1. H; m; Á/ endowed with unital algebra homomorphisms  W H ! H ˝ H , " W H ! C and a linear map S W H ! 22) is called a Hopf algebra. We call  the comultiplication, " the counit, and S the antipode of H . If existence of an antipode is not assumed, then we say we have a bialgebra.

Irreducible representations of G are all 1-dimensional and are parameterized by integers n 2 Z. un / D u n for all n 2 Z. 2/ is the group of unitary 2 by 2 complex matrices with determinant 1. z1 ; z2 / 2 C2 I jz1 j2 Cjz2 j2 D 1g. z1 ; z2 / D z2 . They satisfy the relation ˛˛ C ˇˇ D 1. , U U D U U D I . 2/ are tensor products of the fundamental representation whose matrix is U [21]. 2// generated by ˛ and ˇ. Its coproduct, counit, and antipode are uniquely induced by their values on the generators:  à  à  à ˛ ˇ ˛ ˇ .

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