Quaternion-Octonion SU(3) Flavor Symmetry

Starting with the quaternionic formulation of isospin SU(2) group, we have derived the relations for different components of isospin with quark states. Extending this formalism to the case of SU(3) group, we have considered the theory of octonion variables. Accordingly, the octonion splitting of SU(3) group have been reconsidered and various commutation relations for SU(3) group and its shift operators are also derived and verified for different isospin multiplets i.e. I, U and V-spins.     

Novel oxidatively removable protecting groups and linkers for solid-phase synthesis of oligosaccharides

Summary Several new para-substituted benzyl- or phenyl-type protecting groups and their application to linkers for solid-phase synthesis are described.p-Acylaminobenzyl groups have higher acid stability than thep-methoxybenzyl (MPM) group, but are readily cleaved with 2,3-dichloro-5,6-dicyanobenzoquinone (DDQ). Thep-azidobenzyl (Azb) group also has higher acid stability than the MPM group and can be removed much faster than the MPM group by DDQ oxidation after conversion of the azide group into the corresponding iminophosphorane. The acid stability of thep-azido-m-chlorobenzyl group (Cl-Azb) is higher than that of the Azb group. The former can be readily removed by DDQ oxidation after conversion of the azide group into the iminophosphorane. Thep-acylaminophenyl glycoside linker can be readily obtained fromp-nitrophenyl glycoside and can be readily cleaved by ammonium cerium(IV) nitrate (CAN) oxidation. This type of linker should be useful not only for the solid-phase synthesis of oligosaccharides but also for general solid-phase synthesis.     

The Donald–Flanigan Problem for Finite Reflection Groups

The Donald–Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra kH to a separable algebra. It is solved here for dihedral groups and Weyl groups of types B _n and D _n (whose rational group algebras are computed), leaving but six finite reflection groups with solutions unknown. We determine the structure of a wreath product of a group with a sum of central separable algebras and show that if there is a solution for H over k which is a sum of central separable algebras and if S _n is the symmetric group then (i) the problem is solvable also for the wreath product H S _n = H × ··· × H (n times) S _n and (ii) given a morphism from a finite Abelian or dihedral group G to S _n it is solvable also for H G. The theorems suggested by the Donald–Flanigan conjecture and subsequently proven follow, we also show, from a geometric conjecture which although weaker for groups applies to a broader class of algebras than group algebras.     

Operator-valued connections,Lie connections,and Gauge field theory

Noether's first theorem tells us that the global symmetry groupG _r of an action integral is a Lie group of point transformations that acts on the Cartesian product of the space-time manifold with the space of states and their derivatives. Gauge theory constructs are thus required for symmetry groups that act indiscriminately on the independent and dependent variables where the group structure can not necessarily be realized as a subgroup of the general linear group. Noting that the Lie algebra of a general symmetry groupG _r can be realized as a Lie algebrag _r of Lie derivatives on an appropriately structured manifold,G _r -covariant derivatives are introduced through study of connection 1-forms that take their values in the Lie algebrag _r of Lie derivatives (operator-valued connections). This leads to a general theory of operator-valued curvature 2-forms and to the important special class of Lie connections. The latter are naturally associated with the minimal replacement and minimal coupling constructs of gauge theory when the symmetry groupG _r is allowed to act locally. Lie connections give rise to the gauge fields that compensate for the local action ofG _r in a natural way. All governing field equations and their integrability conditions are derived for an arbitrary finite dimensional Lie group of symmetries. The case whereG _r contains the ten-parameter Poincaré group on a flat space-timeM ₄ is considered. The Lorentz structure ofM ₄ is shown to give a pseudo-Riemannian structure of signature 2 under the minimal replacement associated with the Lie connection of the local action of the Poincaré group. Field equations for the matter fields and the gauge fields are given for any system of matter fields whose action integral is invariant under the global action of the Poincaré group.     

Hamilton's Turns for the Lorentz Group

Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by “turns,” which are equivalence classes of directed great circle arcs on the unit sphere S ², in such a manner that the rule for composition of group elements takes the form of the familiar parallelogram law for the Euclidean translation group. It is only recently that this construction has been generalized to the simplest noncompact group SU(1, 1)=Sp(2, R)=SL(2, R), the double cover of SO(2, 1). The present work develops a theory of turns for SL(2, C), the double and universal cover of SO(3, 1) and SO(3, C), rendering a geometric representation in the spirit of Hamilton available for all low dimensional semisimple Lie groups of interest in physics. The geometric construction is illustrated through application to polar decomposition, and to the composition of Lorentz boosts and the resulting Wigner or Thomas rotation. PACS numbers: 02.20.-a     

Chirality and Symmetry Breaking in a Discrete Internal Space

In previous papers the permutation group S ₄ has been suggested as an ordering scheme for quarks and leptons, and the appearance of this finite symmetry group was taken as indication for the existence of a discrete inner symmetry space underlying elementary particle interactions. Here it is pointed out that a more suitable choice than the tetrahedral group S ₄ is the pyritohedral group A ₄×Z ₂ because its vibrational spectrum exhibits exactly the mass multiplet structure of the 3 fermion generations. Furthermore it is noted that the same structure can also be obtained from a primordial symmetry breaking S ₄→A ₄. Since A ₄ is a chiral group, while S ₄ is achiral, an argument can be given why the chirality of the inner pyritohedral symmetry leads to parity violation of the weak interactions.     

On Lie groups with left-invariant symplectic or Kählerian structures

Left-invariant symplectic structure on a group G; properties of the corresponding Lie algebra g. A unimodular symplectic Lie algebra has to be solvable (see [1]). Symplectic subgroups and left-invariant Poisson structures on a group. Affine Poisson structures: an affine Poisson structure associated to g and admitting g ^* as a unique leaf corresponds to a unimodular symplectic Lie algebra and the associate group is right-affine. If G is unimodular and endowed with a left-invariant metric g, harmonic theory for the left-invariant forms. Kählerian group is metabelian and Riemannianly flat. Decomposition of a simply connected Kählerian group. A symplectic group admitting a left-invariant metric with a nonnegative Ricci curvature is unimodular and admits a left-invariant flat Kählerian structure.     

Generalized coupling constants for broken gauge symmetries in geometrical terms

A geometrical treatment of the gauge coupling constant is proposed in terms of a generalized connection form using fibre-bundle language. This extends the notion of the coupling constant to a notion of a field. The reduction of a curvature form for the generalized connection form is described in the case of a reduction of a structure group G to a subgroup H (broken gauge symmetry), and a coupling constant for the gauge group H is constructed from the corresponding one for the gauge group G.     

A (<Emphasis Type="Italic">t</Emphasis>,<Emphasis Type="Italic">n</Emphasis>)-Threshold Scheme of Multi-party Quantum Group Signature with Irregular Quantum Fourier Transform

A novel (t,n)-threshold scheme for the multi-party quantum group signature is proposed based on the irregular quantum Fourier transform, in which every t-qubit quantum message needs n participants to generate the quantum group signature. All the quantum operation gates in the quantum circuit can be distributed and arranged randomly in the irregular QFT algorithm, which can increase the von Neumann entropy of the signed quantum message and the randomicity of the quantum signature generation significantly. The generation and verification of the quantum group signature can be both performed in quantum circuits with the parallel algorithm. Security analysis shows that an available and legal quantum (t,n)-threshold group signature can be achieved.     

Regular Weyl-Systems and Smooth Structures on Heisenberg Groups

We study representation theory of the Weyl relations for infinitely many degrees of freedom. Differentiability of regular representations along rays in the parameter space E suggests to consider smooth structures on E. Switching from representations of CCR to group representations of the associated Heisenberg group over E we develop a framework for smooth representations of the Heisenberg group as an infinite dimensional Lie group. After careful inspection and translation of the necessary differential geometric input for Kirillov's orbit method we are able to construct a large class of smooth representations. These reproduce the Schr?dinger representation if E is finite dimensional. Received: 10 May 1996 / Accepted: 30 July 1996     

Overall values of conventional discrepancy indices for related and unrelated models of a non-centrosymmetric crystal with a centrosymmetric group

Theoretical expressions for the overall values of the conventional discrepancy indicesR(F) andR(I) are derived for a non-centrosymmetric crystal with a centrosymmetric group by taking the centrosymmetric group and a part of the other atoms in the unit cell as the trial structure. These results are used to obtain tables of values of these indices in terms of the parameter σ _1c ² and σ ₁ ² which define the fractional contribution to the local mean intensity from the centrosymmetric group and all the known atoms respectively. Contribution No. 561     

Lyapunov Bounds for Lattice Gauge Dynamics

We study the classical Hamiltonian dynamics of the Kogut–Susskind model for lattice gauge theories on a finite box in a d-dimensional integer lattice. The coupling constant for the plaquette interaction is denoted λ². When the gauge group is a real or a complex subgroup of a unitary matrix group U(N), N≥ 1, we show that the maximal Lyapunov exponent is bounded by , uniformly in the size of the lattice, the energy of the system as well as the order, N, of the gauge group. Received: 20 December 1997 / Accepted: 21 July 1998     

The multi-dimensional <Emphasis Type="Italic">q</Emphasis>-deformed bosonic Newton oscillator and its inhomogeneous quantum invariance group

We obtain the inhomogeneous invariance quantum group for the multi-dimensional q-deformed bosonic Newton oscillator algebra. The homogenous part of this quantum group is given by the multiparameter quantum group $ GL_{X;q_{ij} } $ GL_{X;q_{ij} } of Schirrmacher where q _ij’s take some special values. We find the R-matrix which gives the non-commuting structure of the quantum group for the two dimensional case.     

Test groups and effect algebras

A test group is a pair (G, T) whereG is a partially ordered Abelian group andT is a generative antichain in its positive cone. It is shown here that effect algebras and algebraic test groups are coextensive, and a method for calculating the algebraic closure of a test group is developed. Some computational algorithms for studying finite effect algebras are introduced, and the problem of finding quotients of effect algebras is discussed.     

Molecular invariants: Atomic-group valence

Summary In this work, a generalized valence is defined for an atomic group; it reduces to the known expressions for the case of an atom in a molecule. It is the sum of the correlations between the fluctuations of the atomic chargesq _C andq _D (C belongs to the group andD does not) around their average values. Numerical results agree with chemical expectation.     

Invariance quantum group of the fermionic oscillator

The fermionic oscillator defined by the algebraic relations cc ^* +c ^* c = 1 and c ² = 0 admits the homogeneous group O(2) as its invariance group. We show that the structure of the inhomogeneus invariance group of this oscillator is a quantum group. Received: 15 July 2002 / Revised version: 14 October 2002 / Published online: 19 February 2003     

Modularity in Orbifold Theory for Vertex Operator Superalgebras

This paper is about the orbifold theory for vertex operator superalgebras. Given a vertex operator superalgebra V and a finite automorphism group G of V, we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under the C₂-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C₂-cofinite and g-rational for any g ∈ G.Supported by NSF grants, China NSF grant 10328102 and a Faculty research grant from the University of California at Santa Cruz     

Quantum mechanics on Riemannian manifold in Schwinger's quantization approach II

The extended Schwinger quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold M is a homogeneous Riemannian space with the given action of an isometry transformation group. Using the identification of M with the quotient space G/H, where H is the isotropy group of an arbitrary fixed point of M, we show that quantum mechanics on G/H possesses a gauge structure, described by a gauge potential that is the connection 1-form of the principal fiber bundle G(G/H, H). The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of the states are developed. Received: 27 June 2000 / Revised version: 10 May 2001 / Published online: 19 July 2001     

Three-Manifold Invariants and Their Relation with the Fundamental Group

We consider the 3-manifold invariant I(M) which is defined by means of the Chern–Simons quantum field theory and which coincides with the Reshetikhin–Turaev invariant. We present some arguments and numerical results supporting the conjecture that for nonvanishing I(M), the absolute value |I(M)| only depends on the fundamental group π₁ (M) of the manifold M. For lens spaces, the conjecture is proved when the gauge group is SU(2). In the case in which the gauge group is SU(3), we present numerical computations confirming the conjecture. Received: 15 November 1996 / Accepted: 17 June 1997