Algebra Abelian

In the algebraic, group-theoretical treatments of non-Abelian systems [66,67-70,77-80] the NACT is usually written in a decomposed form as... 

In this particular case, the algebra (2.5) is trivial, since the operator Jz obviously commutes with itself. Algebras formed by commuting operators are called Abelian. 

As mentioned already in Chapter 2, the algebras U(l) and 0(2) are isomorphic (and Abelian). A consequence of this statement is that in one-dimension there is a large number of potentials that correspond exactly to an algebraic structure with a dynamical symmetry. Of particular interest in molecular physics are ... 

An algebra (or subalgebra) is said to be Abelian if all its elements commute... 

Suppose k is an algebraically closed field of characteristic p > 0 and suppose x Spec(fc) — Ag>5 is a geometric point given by the polarized abelian variety (X0, A0) over Spec (A ). In [Cr] the complete local ring of Agtd at x is computed. We will describe the result. 

Proof. Let 2 C Ag, Fp be the locally closed substack which is the analog of 2 in the case of principally polarized abelian schemes. We use the fact that the algebraic stack 2 is geometrically irreducible (see [EO]). 

No2] P. Norman, Intersections of the components of the moduli space of abelian varieties. Journal of Pure and Applied Algebra 13 (1978), 105-107. 

Oo] F. Oort, Finite group schemes, local moduli for abelian varieties and lifting problems. In Algebraic Geometry, Oslo, 1970, 223-254. Also in Compositio Math. 23 (1972), 265-296. 

The first two chapters are about moduli of abelian varieties, i.e. about classifying abelian varieties. An abelian variety is a proper variety endowed with a group structure. It turns out that any abelian variety A is projective there exists some ample line bundle on A. An ample line bundle determines an isogeny A A — A, which depends on C only up to algebraic equivalence. Such a morphism A is called a polarization, it is called a principal polarization if A is an isomorphism. 

Let (A, A) be a principally polarized abelian variety over an algebraically closed field k. If the characteristic of k is not equal to the prime p, then the kernel of multiplication by p on A(k) is a finite group isomorphic to (Z/pZ)29. The polarization A induces a nondegenerate alternating pairing A k)[p] x A( )[p] -+ pp(k). Hence, we can try to classify principally polarized abelian varieties with a symplectic basis for the group of points of order p. However, this no longer works in characteristic p. 

In this article we study the moduli spaces Ag,d over Spec(Z) parametrizing abelian varieties of dimension g with a polarization of degree d2. We use the language of algebraic stacks since it seems the most natural language to describe moduli spaces with. 

Construction 1.9. We need the algebraic stacks A jl)7l parametrizing abelian schemes with a principal polarization and a sympletic level-n structure . The objects of the underlying category of A >Xyn are quadruples (S,X, A, a) where... 

Let (A, A) be a principally polarized abelian variety of dimension g. In this article a r<)(p)-level structure on (A, A) is a flag of subgroup schemes 0 C Hi C Hg C A p] such that the order of Hi is p and such that Hg is totally isotropic for the Weil-pairing induced by A on A[p. We let S(g,p) be the algebraic stack over Spec(Z) parametrizing principally polarized abelian varieties endowed with a Fo(p)-level structure (Definition 1.1). 

The two parts of the twisted bundle are copies of SU(2) with a doublet fermion structure. One of the fermions has a very large mass, m = Yn (y)1 ), which is assumed to be unstable and not observed at low energies. So one sector of the twisted bundle is left with the same Abelian structure, but with a singlet fermion, meaning that the SU(2) gauge theory becomes defined by the algebra over the basis elements... 

Unfortunately, it is not possible to automatically generalize the Abelian Stokes theorem [e.g., Eq. (4)] to the non-Abelian one. In the non-Abelian case one faces a qualitatively different situation because the integrand on the l.h.s. assumes values in a Lie algebra g rather than in the field of real or complex numbers. The picture simplifies significantly if one switches from the local language to a global one [see Eq. (5)]. Therefore we should consider the holonomy (7) around a closed curve C ... 

As this concerns the nature of non-Abelian electrodynamics, we will pursue the matter of a GUT that incorporates non-Abelian electrodynamics. This GUT will be an 50(10) theory as outlined above. We have that an extended electro-weak theory that encompasses non-Abelian electrodynamics is spin(4) = 51/(2) x 517(2). This in turn can be embedded into a larger 50(10) algebra with spin(6) = 517(4). 50(10) may be decomposed into 517(2) x 517(2) x 517(4). This permits the embedding of the extended electro weak theory with 517(4), which may contain the nuclear interactions as 517(4) 51/(3) x 1/(1). In the following paragraphs we will discuss the nature of this gauge theory and illustrate some basic results and predictions on how nature should appear. We will also discuss the nature of fermion fields in an 517(2) x 51/(2) x 51/(4) theory. 

Algebraic Invariants of Knots and Links, and Non-Abelian Field Models... 

Theorem. Let k be perfect, S an abelian algebraic matrix group. Let Ss and Su be the sets of separable and unipotent elements in S. Then Ss and Su are closed subgroups, and S is their direct product. 

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