Elements Of SL Research Articles

Spinor extended Lorentz-force-like equations as consequence of a spinorial structure of space-time

As previously shown, the special relativistic dynamical equation of the Lorentz force type can be regarded as a consequence of a succession of space-time dependent infinitesimal Lorentz boosts and rotations. This insight indicates that the Lorentz-Force-like equation has a fundamental meaning in physics. We show how this result may be spinorially obtained starting out from the application of an infinitesimal element of SL(2,C) to the individual spinors, which are regarded here as being more fundamental objects than four-vectors. In this way we get a set of new dynamical spinor equations inducing the extended Lorentz-Force-like equation in the Minkowski space-time and geometrically obtain the spinor form of the electromagnetic field tensor. The term extended refers to the dynamics of some additional degrees of freedom that may be associated with the intrinsic spin, namely, with the dynamics of three spacelike mutually orthogonal four-vectors, all of them orthogonal to the linear four-momentum of the object under consideration that finally, in the particle’s proper frame, are identified with the generators of SU(2).

Diameters of Cayley graphs of Chevalley groups

We show that for integers k ≥ 2 and n ≥ 3 , the diameter of the Cayley graph of SL n ( Z / k Z ) associated with a standard two-element generating set is at most a constant times n 2 ln k . This answers a question of A. Lubotzky concerning SL n ( F p ) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing elements of SL n ( Z / k Z ) . We generalize our results to other Chevalley groups over Z / k Z .

Order of elements in the groups related to the general linear group

For a natural number n and a prime power q the general, special, projective general and projective special linear groups are denoted by GL n ( q ) , SL n ( q ) , PGL n ( q ) and PSL n ( q ) , respectively. Using conjugacy classes of elements in GL n ( q ) in terms of irreducible polynomials over the finite field GF ( q ) we demonstrate how the set of order elements in GL n ( q ) can be obtained. This will help to find the order of elements in the groups SL n ( q ) , PGL n ( q ) and PSL n ( q ) . We also show an upper bound for the order of elements in SL n ( q ) .

Isotropic nonarchimedean S-arithmetic groups are not left orderable

If O S is the ring of S-integers of an algebraic number field F, and O S has infinitely many units, we show that no finite-index subgroup of SL ( 2 , O S ) is left orderable. (Equivalently, these subgroups have no nontrivial orientation-preserving actions on the real line.) This implies that if G is an isotropic F-simple algebraic group over an algebraic number field F, then no nonarchimedean S-arithmetic subgroup of G is left orderable. Our proofs are based on the fact, proved by D. Carter, G. Keller, and E. Paige, that every element of SL ( 2 , O S ) is a product of a bounded number of elementary matrices. To cite this article: L. Lifschitz, D.W. Morris, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Regular Elements and Global Controllability in SL(d, R)

We introduce a generalized concept of split-regular elements and study the dynamics of such elements on the flag manifolds corresponding to the Lie group SL(d, R). Based on these concepts, we obtain the stable and unstable manifolds, the Morse decomposition, and obtain the limit sets inside the corresponding control sets. These dynamical aspects are studied with the purpose of applying them in order to extend a result of global controllability of discrete-time bilinear systems defined by elements of sl(d, R).

A new parametrization and all integral realizations of the Lorentz group

A simple linear transformation of the biquaternionic parameters α and β of the group SL(2, C), which is two-to-one homomorphic to the restricted Lorentz group ℒ, is used to express each element of SL(2, C) and ℒ in terms of the first column of an element of ℒ and the quaternionic parameters α. This parametrization is shown to bring the determination of the integral realizations of the restricted Lorentz group under the purview of classical Diophantine analysis, involving the expression of a given pair of integers, respectively, as a sum of three squares and a sum of four squares. These solutions are further constrained by the condition that a set of four integers, which is linear in the solution of the three squares problem and in the solution of the four squares problem, have a common factor. These are all classical problems addressed and solved by Euclid, Euler, Fermat, Gauss, Jacobi, and others. The corresponding realizations of SL(2, C) fall into three distinct classes: those in which the elements of the matrix are Gaussian integers, Gaussian integers divided by √, and Gaussian half-odd integers. These results apply also to the principal subgroups of SL(2, C).

The conformal boundary states for SU(2) at level 1

For the case of the SU(2) WZW model at level one, the boundary states that only preserve the conformal symmetry are analysed. Under the assumption that marginal deformations of the usual Cardy boundary states are consistent, the most general conformal boundary states are determined. They are found to be parametrised by group elements in SL (2, C) .

A Geometric Interpretation of the χ y Genus¶on Hyper-Kähler Manifolds

The group SL(2) acts on the space of cohomology groups of any hyper-Kahler manifold X. The \chi_{y} genus of a hyper-Kahler X is shown to have a geometric interpretation as the super trace of an element of SL(2). As a by product one learns that the generalized Casson invariant for a mapping torus is essentially the \chi_{y} genus.

SL (3, C ) generator matrix elements in a Pauli subgroup basis

We construct SL(3,C) basis states reduced according to its finite subgroup ℘3. Matrix elements of sl(3,C) generators are calculated between ℘3 basis states.

Möbius Type Semigroups in SL(2,)

with an order P, the semigroup S of elements in SL(2,F) having entries in P can be factored into upper triangular, diagonal and lower triangular matrices. Moreover, we show that the semigroup ±S is maximal in SL(2,F).

An algorithm to compute bases and representation matrices for SLn + 1 -representations

We construct a basis for irreducible representations of the complex Lie algebra sL n + 1 . The basis is obtained by applying certain monomials in the enveloping algebra of SL n + 1 to a highest weight vector. In addition we provide a straightening law which can be used to define an algorithm to compute the representation matrix of elements of sL n + 1 with respect to this basis. The method can be generalized to all complex simple Lie algebras with a simply laced root system.

Representation of an element in SL(2, ,) as a short word in two generators

It is one of the first theorems proved in the theory of modular forms [1, p. 78] that every element in SL (2, ℤ), and hence in SL(2, ), can be written as a product of the two generatorsWhile this is quite an easy result, it required more advanced tools such as spectral analysis on Riemann surfaces [2] to show that there is in fact always such a product of length O(log p). This result is not constructive and therefore immediately [2, p. 102] raises the following:Problem. Is there an algorithm polynomial in log p for constructing a monomial in S and T, of degree O(log p), whose value isIn this note, we construct such an algorithm. We will carry out all the details only for the particular elementbut it will become obvious that our method works for all elements in SL(2, ).

Products of similar matrices

Given a nontrivial conjugacy class Ω of GL( V) which is contained in SL( V), we want to find a minimal integer k such that every element of SL( V) is a product of at most k elements of Ω. The analogous question is studied when Ω is replaced by Ф · Ф −1 , where Ф is a nontrivial conjugacy class of GL( V).

VOLKOV'S ALGEBRA AND CASIMIR ELEMENT OF SL2

We construct 3-D module of sl 2 and then obtain the generator of classical Volkov's algebra as Casimir element of sl 2. Only classical level is discussed.

Efficient Parallel k-Set Chain Range-Join in Hypercubes

The chain range-join of k sets, St,S2, ,Sk, is the set containing all tuples (st,S2, • • •,s*) that satisfy «,•'' ) local memory at each processor, and has a time complexity at most O(((nk/p) + nk_i)log(nk/p)) in the best case when no element in Sl+i matches any element in S, for 1 ^ t < k — 1, O(kTsort + (k 2 /p) n?=i;) m tn e worst case when all elements in Sl+i match each element in S,, where Tson = #((£//') 11?= 2 n il°gn?= 2i) when all elements in S,+ 1 are distinct, and Tson = O((k/p) n?=2i) when all elements in S,+i are equal. The general-case time complexity of the algorithm is also shown. The algorithm is implemented on a UNIX-based network using a simulator designed in C and its performance is fully evaluated through extensive testing.

Generating Random Elements in SL n (F q ) by Random Transvections

This paper studies a random walk based on random transvections in SL n(F q ) and shows that, given \(\in\) > 0, there is a constant c such that after n + c steps the walk is within a distance \(\in\) from uniform and that after n − c steps the walk is a distance at least 1 − \(\in\) from uniform. This paper uses results of Diaconis and Shahshahani to get the upper bound, uses results of Rudvalis to get the lower bound, and briefly considers some other random walks on SL n(F q ) to compare them with random transvections.

Functions of two-dimensional Bravais lattices

The symmetries of a function are considered that takes as its argument an arbitrary two-dimensional Bravais lattice, represented by a 2×2 matrix. Rotating the lattice corresponds to multiplying its matrix on the right by an element of SO(2). Matrices related by left multiplication of an element of SL(2,Z) also refer to the same lattice. Functions of matrices having these two symmetries possess an expansion known as the Roelcke–Selberg decomposition. This decomposition characterizes the function in terms of a set of coefficients (the discrete spectrum) and a function defined on a line in the complex plane (the continuous spectrum). When the function of the lattice refers to its energy, the continuous spectrum can be related to an interatomic potential between the atoms of the lattice. The decomposition can be considered a Landau theory of a strained two-dimensional Bravais lattice, valid for arbitrarily large strains, expanded about any structure. This type of theory may be useful in determining the energies of a material for many different lattice configurations from a knowledge of the energy at only a few of them.

Generalized Apollonian packings

In this paper we generalize the classical two-dimensional Apollonian packing of circles to the case where the circles are no more tangent. We introduce two elements ofSL(2,ℂ) as generators:R andT that are hyperbolic rotations of 2π/3 and 2π/N (N=2,3,4....), around two distinct points. The limit set of the discrete group generated byR andT provides, forN=7,8,9,.... a generalization of the Apollonian packing (which is itself recovered forN=∞). The valuesN=2,3,4,5 produce a very different result, giving rise to the rotation groups of the cube forN=2 and 4, and the icosahedron forN=3 and 5. ForN=6 the group is no longer discrete. To further analyze this structure forN≥7, we move to the Minkowski space in which the group acts on a one sheeted hyperboloid. The circles are now represented by points on this variety and generate a crystal on it.

Some subgroups of SL(3, z) generated by involutions

For R a commutative ring with identity 1 we let SL(n, R) denote the group of n × n integral matrices with determinant 1. A transvection T is an element of SL(n, R) which we represent (see [1]) as a pair (φ d) where φ ∈ (Rn)*, the dual space of Rn, d ∈ Rn, φ(d) = 0, and for all x ∈ Rn we haveT(x) = + φ(x) d.Throughout this paper an involution is an element Y of SL(n, R) which has order two. Let n = 3 and R = Z and let C = diag(–1, –1, –1) be the central element of GL(3, Z).

Minimal parabolic terms in the Selberg trace formula for SL(3, [formula omitted])\\ SL(3, [formula omitted])/ SO(3, [formula omitted

In this paper we compute orbital integrals occurring in the Selberg trace formula for SL(3, Z )\\ SL(3, R )/ SO(3, R ) which are associated to unipotent elements of SL(3, Z ). An effort is made to reduce the truncated integrals to the lower rank case whenever possible because this phenomenon will cause badly divergent parts of the formula to disappear later. Those integrals which do not reduce to a lower rank case are also described.