Minimal spin Gauge theory

Purpose of a minimal theory is to achief most with least. Least may be for example the spacetime algebra. But the symmetric unitary groupSU(3) is not a part of any real Clifford algebra of 4-dimensional space, especially not of the algebraCl _1,3 of the Minkowski spacetime, nor of the algebraCl _3,1 in the opposite metric. Therefore we can ask how quantumchromodynamics enters into the theory. A first answer is that the groupSU(3) is an object of both the complexified algebras C ⊕Cl _1,3 and C⊕Cl _3,1. To show this we first define six color spaces which are spanned by conjugate triples of commuting base elements. These contain the six idempotent lattices that can be located inCl _3,1. Their images exist in both C⊕Cl _1,3 and C⊕Cl _3,1. Further in each color space there is defined an octahedral orientation stabilizer group which fixates one lepton and color rotates the states in its quark family. Thus quantum numbers of strong interacting fields such as isospin, charge, hypercharge and color turn out as geometric properties. Next we ask if the artificialty of complexification can be avoided. The answer is yes. Defining the class of Clifford algebras with proper imaginary unit it turns out thatCl _1,3 andCl _3,1 do not belong to this class. ButCl _4,1 andCl _1,6 do. It is shown that in the latter algebra the whole color space Ansatz can be established and the generators ofSU(3) represented most naturally and without complexification. That the proposed theory becomes a physically true statement requires that there exists a non rank preserving freedom of motion within the constituents of primitive idempotents, that is, transpositions among conjugate triples in color space.     

Finite-Range Electromagnetic Interaction and Magnetic Charges Spacetime Algebra or Algebra of Physical Space?

A finite-range electromagnetic (EM) theory containing both electric and magnetic charges constructed using two vector potentials A^μ and Z^μ is formulated in the spacetime algebra (STA) and in the algebra of the three-dimensional physical space (APS) formalisms. Lorentz, local gauge and EM duality invariances are discussed in detail in the APS formalism. Moreover, considerations about signature and dimensionality of spacetime are discussed. Finally, the two formulations are compared. STA and APS are equally powerful in formulating our model, but the presence of a global commuting unit pseudoscalar in the APS formulation and the consequent possibility of providing a geometric interpretation for the imaginary unit employed throughout physics lead us to prefer the APS approach.     

Symmetric spaces with maximal projection constants

In this paper we introduce a special class of finite-dimensional symmetric subspaces of L₁, so-called regular symmetric subspaces. Using this notion, we show that for any k?2, there exist k-dimensional symmetric subspaces of L₁ which have maximal projection constant among all k-dimensional symmetric spaces. Moreover, L₁ is a maximal overspace for these spaces (see Theorems 4.4 and 4.5.) Also a new asymptotic lower bound for projection constants of symmetric spaces is obtained (see Theorem 5.3). This result answers the question posed in [12, p. 36] (see also [15, p. 38]) by H. Koenig and co-authors. The above results are presented both in real and complex cases.     

Algorithms for Computations in Local Symmetric Spaces

In the last two decades much of the algebraic/combinatorial structure of Lie groups, Lie algebras, and their representations has been implemented in several excellent computer algebra packages, including LiE, GAP4, Chevie, Magma, and Maple. The structure of reductive symmetric spaces or more generally symmetric k-varieties is very similar to that of the underlying Lie group, with a few additional complications. A computer algebra package enabling one to do computations related to these symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc.

In this article we lay the groundwork for computing the fine structure of symmetric spaces over the real numbers and other base fields, give a complete set of algorithms for computing the fine structure of symmetric varieties and use this to compute nice bases for the local symmetric varieties.     

On the stable rank and reducibility in algebras of real symmetric functions

Let A_?(??) denote the set of functions belonging to the disc algebra having real Fourier coefficients. We show that A_?(??) has Bass and topological stable ranks equal to 2, which settles the conjecture made by Brett Wick in [18]. We also give a necessary and sufficient condition for reducibility in some real algebras of functions on symmetric domains with holes, which is a generalization of the main theorem in [18]. A sufficient topological condition on the symmetric open set ?? is given for the corresponding real algebra A_?(??) to have Bass stable rank equal to 1 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximation Numbers of Identity Operators between Symmetric Sequence Spaces

We prove asymptotic formulas for the behavior of approximation quantities of identity operators between symmetric sequence spaces. These formulas extend recent results of Defant, Masty o, and Michels for identities l_pⁿF_n with an n-dimensional symmetric normed space F_n with p-concavity conditions on F_n and 1p2. We consider the general case of identities E_nF_n with weak assumptions on the asymptotic behavior of the fundamental sequences of the n-dimensional symmetric spaces E_n and F_n. We give applications to Lorentz and Orlicz sequence spaces, again considerably generalizing results of Pietsch, Defant, Masty o, and Michels.     

Representations of the virasoro-like algebra and its q-Analog

In this paper, some irreducible graded modules with 1-dimensional homogeneous spaces over the Virasoro-like algebra and its q-analogs are constructed. The unitarizability of these modules, and the conditions under which two of such irreducible graded modules are ismorphic are determined. Some other kinds of irreducible graded modules with 1-dimensional homogeneous spaces over the Virasorolike algebra and its q-analogs are also given.     

n-Dimensional dual complex numbers

It is well-known that the quadratic algebrasQ _a,b = {z|z =x +qy,q ² =a +qb ,a, b, x, y ε ℝ,q ∉ ℝ }, also expressible as ℝ[x]/(x ² -bx -a), are, up to isomorphism, equivalent to just three algebras, corresponding to elliptic, parabolic and hyperbolic. These three types are usually represented byQ _−1,0,Q _0,0,Q _1,0 and called complex numbers, dual complex numbers and hyperbolic complex numbers, respectively. Each in turn describes a Euclidian, Galilean and Minkowskian plane. The hyperbolic complex numbers thus provide a 2-dimensional spacetime for special relativity physics (see e.g. [6]) and the dual complex numbers a 2-dimensional spacetime for Newtonian physics (see e.g. [17]). The present authors considered extensions of the hyperbolic complex numbers ton dimensions in [8], and here, in somewhat parallel fashion, some elements of algebra (in Section 1) and analysis (in Section 2) will be presented forn-dimensional dual complex numbers.      

Model-theory of vector-spaces over unspecified fields

Vector spaces over unspecified fields can be axiomatized as one-sorted structures, namely, abelian groups with the relation of parallelism. Parallelism is binary linear dependence. When equipped with the n-ary relation of linear dependence for some positive integer n, a vector-space is existentially closed if and only if it is n-dimensional over an algebraically closed field. In the signature with an n-ary predicate for linear dependence for each positive integer n, the theory of infinite-dimensional vector spaces over algebraically closed fields is the model-completion of the theory of vector spaces.      

Twistors, 4-symmetric spaces and integrable systems

An order four automorphism of a Lie algebra gives rise to an integrable system introduced by Terng. We show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a Riemannian symmetric space. As applications, we find that surfaces with holomorphic mean curvature in 4-dimensional real or complex space forms constitute an integrable system as do Hamiltonian stationary Lagrangian surfaces in 4-dimensional Hermitian symmetric spaces (this last providing a conceptual explanation of a result of Hélein-Romon).     

POLYNOMIAL IDENTITIES ON SUPERALGEBRAS AND ALMOST POLYNOMIAL GROWTH

Let A be a superalgebra over a field of characteristic zero. In this paper we investigate the graded polynomial identities of A through the asymptotic behavior of a numerical sequence called the sequence of graded codimensions of A. Our main result says that such sequence is polynomially bounded if and only if the variety of superalgebras generated by A does not contain a list of five superalgebras consisting of a 2-dimensional algebra, the infinite dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with trivial and nontrivial gradings. Our main tool is the representation theory of the symmetric group.     

Codimension of immersions with parallel pluri-mean curvature

Immersions with parallel pluri-mean curvature into euclidean n-space generalize constant mean curvature immersions of surfaces to Kähler manifolds of complex dimension m. Examples are the standard embeddings of Kähler symmetric spaces into the Lie algebra of its transvection group. We give a lower bound for the codimension of arbitrary ppmc immersions. In particular we show that M is locally symmetric if the codimension is minimal.     

On the transposition anti-involution in real Clifford algebras I: the transposition map

A particular orthogonal map on a finite-dimensional real quadratic vector space (V,?Q) with a non-degenerate quadratic form Q of any signature (p,?q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra C?(V*,?Q) of linear functionals (multiforms) acting on the universal Clifford algebra C?(V,?Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of C?(V,?Q). The anti-involution reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the element matrix in the left regular representation of C?(V,?Q). We also give an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [R. Ab?amowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Algebra, to appear].     

q-Difference intertwining operators and q-conformal invariant equations

We first recall a canonical procedure for the construction of the invariant differential operators and equations for arbitrary complex or real noncompact semisimple Lie groups. Then we present the application of this procedure to the case of quantum groups. In detail is given the construction of representations of the quantum algebra U _q (sl(n)) labelled by n–1 complex numbers and acting in the spaces of functions of n(n–1)/2 noncommuting variables, which generate a q-deformed SL(4) flag manifold. The conditions for reducibility of these representations and the procedure for the construction of the q-difference intertwining operators are given. Using these results for the case n=4 we propose infinite hierarchies of q-difference equations which are q-conformal invariant. The lowest member of one of these hierarchies are new q-Maxwell equations. We propose also new q-Minkowski spacetime which is part of a q-deformed SU(2,2) flag manifold.     

Produkte positiver elemente in formal-reellen Jordan-Algebren

A generalization of the Rayleigh quotient defined for real symmetric matrices to the elements of a formally real Jordan algebra is used here to give a generalization to formally real Jordan algebras of the theorem that for any real symmetric matrix C with tr C > 0 there are positive definite real symmetric matrices A and B with C = AB + BA.     

Quantum octonions

This paper constructs a quantum deformation of the complex Cayley dgebra. The method uses the representation theory of U _q(sl(2)), the quantized enveloping algebra of the simple complex Lie algebra s/(2). The paper begins by constructing a quantum deforma-tion of the complex quaternion algebra, since this simpler case illustrates all of the necessary steps. As intermediate results, deformations are constructed of sl(2) and the 7-dimensional simple Malcev algebra.     

Homogeneous nilmanifolds attached to representations of compact Lie groups

For each compact Lie algebra ? and each real representation V of ? we consider a two-step nilpotent Lie group N(?,V), endowed with a natural left-invariant riemannian metric. The homogeneous nilmanifolds so obtained are precisely those which are naturally reductive. We study some geometric aspects of these manifolds, finding many parallels with H-type groups. We also obtain, within the class of manifolds N(?,V), the first examples of non-weakly symmetric, naturally reductive spaces and new examples of non-commutative naturally reductive spaces. Received: 16 September 1998 / Revised version: 24 February 1999     

Riemannian Manifolds in Which Certain Curvature Operator Has Constant Eigenvalues along Each Circle

Riemannian manifolds for which a natural curvature operator has constant eigenvalues on circles are studied. A local classification in dimensions two and three is given. In the 3-dimensional case one gets all locally symmetric spaces and all Riemannian manifolds with the constant principal Ricci curvatures r ₁ = r ₂ = 0, r ₃= 0 , which are not locally homogeneous, in general.