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We discuss the existence of an orthogonal basis consisting of decomposable vectors for some symmetry classes of tensors associated with Semi-Dihedral groups $SD_{8n}$. The dimensions of these symmetry classes of tensors are also computed.

Toral automorphisms are widely used (discrete) dynamical systems, the perhaps most prominent example (in 2D) being Arnold's cat map. Given such an automorphism M, its symmetries (i.e. all automorphisms that commute with M) and reversing symmetries (i.e. all automorphisms that conjugate M into its inverse) can be determined by means of number theoretic tools. Here, the case of GL(2,Z) is presented and the possible (reversing) symmetry groups are completely classified. Extensions to affine mappings and to k-(reversing) symmetries (i.e. (reversing) symmetries of the k-th power of M), and applications to the projective group PGL(2,Z) and to trace maps (compare math.DS/9901124), are briefly discussed.

We present the group-thoretical classification of gap functions in superconductors coexisting with some magnetic order in non-symmorphic magnetic space groups. Based on the weak-coupling BCS theory, we show that UCoGe-type ferromagnetic superconductors must have horizontal line nodes on either $k_z=0$ or $\pm\pi/c$ plane. Moreover, it is likely that additional Weyl point nodes exist at the axial point. On the other hand, in UPd$_2$Al$_3$-type antiferromagnetic superconductors, gap functions with $A_g$ symmetry possess horizontal line nodes in antiferromagnetic Brillouin zone boundary perpendicular to $c$-axis. In other words, the conventional fully-gapped $s$-wave superconductivity is forbidden in this type of antiferromagnetic superconductors, irrelevant to the pairing mechanism, as long as the Fermi surface crosses a zone boundary. UCoGe and UPd$_2$Al$_3$ are candidates for unconventional superconductors possessing hidden symmetry-protected line nodes, peculiar to non-symmorphic magnetic space groups.

We consider amalgamated free product II$_1$ factors $M = M_1 *_B M_2 *_B ...$ and use ``deformation/rigidity'' and ``intertwining'' techniques to prove that any relatively rigid von Neumann subalgebra $Q\subset M$ can be intertwined into one of the $M_i$'s. We apply this to the case $M_i$ are w-rigid II$_1$ factors, with $B$ equal to either $\Bbb C$, to a Cartan subalgebra $A$ in $M_i$, or to a regular hyperfinite II$_1$ subfactor $R$ in $M_i$, to obtain the following type of unique decomposition results, \`a la Bass-Serre: If $M = (N_1 *_C N_2 *_C ...)^t$, for some $t>0$ and some other similar inclusions of algebras $C\subset N_j$ then, after a permutation of indices, $(B\subset M_i)$ is inner conjugate to $(C\subset N_i)^t$, $\forall i$. Taking $B=\Bbb C$ and $M_i = (L(\Bbb Z^2 \rtimes \Bbb F_{2}))^{t_i}$, with $\{t_i\}_{i\geq 1}=S$ a given countable subgroup of $\Bbb R_+^*$, we obtain continuously many non stably isomorphic factors $M$ with fundamental group $\mycal F(M)$ equal to $S$. For $B=A$, we obtain a new class of factors $M$ with unique Cartan subalgebra decomposition, with a large subclass satisfying $\mycal F(M)=\{1\}$ and Out$(M)$ abelian and calculable. Taking $B=R$, we get examples of factors with $\mycal F(M)=\{1\}$, Out$(M)=K$, for any given separable compact abelian group $K$.

In this paper we describe how one can obtain Lie group structures on the group of (vertical) bundle automorphisms for a locally convex principal bundle P over the compact manifold M. This is done by first considering Lie group structures on the group of vertical bundle automorphisms Gau(P). Then the full automorphism group Aut(P) is considered as an extension of the open subgroup Diff(M)_P of diffeomorphisms of M preserving the equivalence class of P under pull-backs, by the gauge group Gau(P). We derive explicit conditions for the extensions of these Lie group structures, show the smoothness of some natural actions and relate our results to affine Kac--Moody algebras and groups.

Much of the recent progress in String Theory can be traced to a precise strategy: a careful study of the few models known since the beginnings of the subject, and the abstraction from them of basic properties that one would like to demand from other models. This could be termed a set of "model-building rules". The approach corresponds to the fact, often a source of embarrassment to specialists, that String Theory, born as a set of rules rather than as a set of principles, has long resisted attempts to reduce it to a logically satisfying structure. Talk presented at the Cargese Summer Institute on Non-Perturbative Methods in Field Theory, Cargese, France, July 16-30, 1987.

For a discrete mechanical system on a Lie group $G$ determined by a (reduced) Lagrangian $\ell$ we define a Poisson structure via the pull-back of the Lie-Poisson structure on the dual of the Lie algebra ${\mathfrak g}^*$ by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group $G$ of the canonical discrete Lagrange 2-form $\omega_\mathbb{L}$ on $G \times G$. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system. Links between our approach and that of groupoids and algebroids as well as the reduced Hamilton-Jacobi equation are made. The rigid body is discussed as an example.

We introduce and study natural two-parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups. Specifically, for each pair (p,q) of non-negative integers we define and investigate quantum groups O^+(p,q), B^+(p,q), S^+(p,q) and H^+(p,q) corresponding to, respectively, orthogonal groups, bistochastic groups, symmetric groups and hyperoctahedral groups. In the first three cases the new quantum groups turn out to be related to the (dual free products of) free quantum groups studied earlier. For H^+(p,q) the situation is different: we show that H^+(p,0) is isomorphic to the quantum isometry group of the C*-algebra of the free group and it can be viewed as a liberation of the classical isometry group of the p-dimensional torus.

A classification of the possible symmetric principal bundles with a compact gauge group, a compact symmetry group and a base manifold which is regularly foliated by the orbits of the symmetry group is derived. A generalization of Wang's theorem (classifying the invariant connections) is proven and local expressions for the gauge potential of an invariant connection are given.

On June 5, 2007 the second author delivered a talk at the Journees de l'Institut Elie Cartan entitled "Finite symmetry groups in complex geometry". This paper begins with an expanded version of that talk which, in the spirit of the Journees, is intended for a wide audience. The later paragraphs are devoted both to the exposition of basic methods, in particular an equivariant minimal model program for surfaces, as well as an outline of recent work of the authors on the classification of K3-surfaces with special symmetry.

If a (possibly finite) compact Lie group acts effectively, locally linearly, and homologically trivially on a closed, simply-connected four-manifold with second Betti number at least three, then it must be isomorphic to a subgroup of S^1 x S^1, and the action must have nonempty fixed-point set. Our results strengthen and complement recent work by Edmonds, Hambleton and Lee, and Wilczynski, among others. Our tools include representation theory, finite group theory, and Borel equivariant cohomology.

The Euler equation (EE) is one of the basic equations in many physical fields such as fluids, plasmas, condensed matter, astrophysics, oceanic and atmospheric dynamics. A symmetry group theorem of the (2+1)-dimensional EE is obtained via a simple direct method which is thus utilized to find \em exact analytical \rm vortex and circumfluence solutions. A weak Darboux transformation theorem of the (2+1)-dimensional EE can be obtained for \em arbitrary spectral parameter \rm from the general symmetry group theorem. \rm Possible applications of the vortex and circumfluence solutions to tropical cyclones, especially Hurricane Katrina 2005, are demonstrated.

One applies the symmetry group theory for study the partial differential equations of Tzitzeica surfaces theory. One finds infinitesimal symmetries, Lagrangians and a new solution of Titzeica equation.

We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the first part of the paper, we show that, under mild restrictions, the moduli spaces of smooth curves with an abelian group of automorphisms of a fixed topological type have finitely generated Picard groups. In certain special cases, we are able to compute them exactly. In the second part of the paper, we show that finite abelian level covers of the hyperelliptic locus in the moduli space of smooth curves have finitely generated Picard groups. We also compute the Picard groups of the moduli spaces of hyperelliptic curves of compact type.

We give a necessary and sufficient condition for the mapping class group of the pair of the 3-sphere and a graph embedded in it to be isomorphic to the topological symmetry group of the embedded graph.

We give a necessary and sufficient condition for the mapping class group of the pair of the 3-sphere and a graph embedded in it to be isomorphic to the topological symmetry group of the embedded graph.

Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the n-body problem: periodic motions where the n bodies all follow one another at regular intervals along a closed path. The principal approach combines variational methods with symmetry properties. In this paper, we give a systematic treatment of the symmetry aspect. In the first part we classify all possible symmetry groups of planar n-body, collision-free choreographies. These symmetry groups fall in to 2 infinite families and, if n is odd, three exceptional groups. In the second part we develop the equivariant fundamental group and use it to determine the topology of the space of loops with a given symmetry, which we show is related to certain cosets of the pure braid group in the full braid group, and to centralizers of elements of the corresponding coset.

A discrete group $\G$ is called rigidly symmetric if for every $C^*$-algebra $\A$ the projective tensor product $\ell^1(\G)\widehat\otimes\A$ is a symmetric Banach $^*$-algebra. For such a group we show that the twisted crossed product $\ell^1_{\alpha,\o}(\G;\A)$ is also a symmetric Banach $^*$-algebra, for every twisted action $(\alpha,\o)$ of $\G$ in a $C^*$-algebra $\A$\,. We extend this property to other types of decay, replacing the $\ell^1$-condition. We also make the connection with certain classes of twisted kernels, used in a theory of integral operators involving group $2$-cocycles. The algebra of these kernels is studied, both in intrinsic and in represented version.

These are lecture notes of a course on symmetry group analysis of differential equations, based mainly on P. J. Olver's book 'Applications of Lie Groups to Differential Equations'. The course starts out with an introduction to the theory of local transformation groups, based on Sussman's theory on the integrability of distributions of non-constant rank. The exposition is self-contained, pre-supposing only basic knowledge in differential geometry and Lie groups.

Let P and Q be convex polyhedra in E3 with face lattices F(P) and F(Q) and symmetry groups G(P) and G(Q), respectively. Then, P and Q are called face equivalent if there is a lattice isomorphism between F(P) and F(Q); P and Q are called symmetry equivalent if the action of G(P) on F(P) is equivalent to the action of G(Q) on F(Q). It is well known that the set [P] of all polyhedra which are face equivalent to P has the structure of a manifold of dimension {e-1}, up to similarities, where e=e(P) is the number of edges of P. This is a consequence of the Steinitz's classical Theorem. We give a new proof of this fact. The symmetry type of P denoted by is the set of all polyhedron Q symmetry equivalent to P. We show that for polyhedra with symmetry group G(P) a reflection group the dimension of this manifold is {O-1} where O is the number of edge orbits of P under and the action of G(P) on F(P).

We prove that under a symmetry assumption all cocycles on Hopf *-algebras arise from generating functionals. This extends earlier results of R.Vergnioux and D.Kyed and has two quantum group applications: all quantum L\'evy processes with symmetric generating functionals decompose into a maximal Gaussian and purely non-Gaussian part and the Haagerup property for discrete quantum groups is characterized by the existence of an arbitrary proper cocycle.

We give a combinatorial characterization of minimally rigid planar frameworks with orientation-preserving crystallographic symmetry, under the constraint of forced symmetry. The main theorems are proved by extending the methods of the first paper in this sequence from groups generated by a single rotation to groups generated by translations and rotations. The proofs make use of a new family of matroids defined on crystallographic groups and associated submodular functions.

We give a combinatorial characterization of minimally rigid planar frameworks with orientation-preserving crystallographic symmetry, under the constraint of forced symmetry. The main theorems are proved by extending the methods of the first paper in this sequence from groups generated by a single rotation to groups generated by translations and rotations. The proofs make use of a new family of matroids defined on crystallographic groups and associated submodular functions.

We analyze D-brane states and their central charges on the resolution of C^2/Z_n by using local mirror symmetry. There is a point in the moduli space where all n(n-1)/2 branches of the principal component of the discriminant locus coincide. We argue that this is the point where compactifications of Type IIA theory on a K3 manifold containing such a local geometry acquire a non-perturbative gauge symmetry of the type A_{n-1}. This analysis, which involves an explicit solution of the GKZ system of the local geometry, explains how the quantum geometry exhibits all positive roots of A_{n-1} and not just the simple roots that manifest themselves as the exceptional curves of the classical geometry. We also make some remarks related to McKay correspondence.

We consider set of functions on Poisson manifold related by continues one-parameter group of transformations. Class of vector fields that produce involutive families of functions is investigated and relationship between these vector fields and non-Noether symmetries of Hamiltonian dynamical systems is outlined. Theory is illustrated with sample models: modified Boussinesq system and Broer-Kaup system.

Let $G$ be a finite group acting linearly on a vector space $V$. For a subset $S\subseteq V$, the linear symmetry group $\operatorname{GL}(S)$ is defined as the set of all linear maps of the linear span of $S$ which permute $S$. We consider the linear symmetry groups $\operatorname{GL}(Gv)$ of orbits $Gv\subseteq V$. We assume that $V$ is the linear span of at least one orbit $Gv$. We define a set of generic points in $V$, which is Zariski-open in $V$, and show that the groups $\operatorname{GL}(Gv)$ for $v$ generic are all isomorphic, and isomorphic to a subgroup of every symmetry group $\operatorname{GL}(Gw)$ such that $V$ is the linear span of $Gw$. If the underlying characteristic is zero, "isomorphic" can be replaced by "conjugate in $\operatorname{GL}(V)$". Moreover, in the characteristic zero case, we show how the character of $G$ on $V$ determines this generic symmetry group. We apply our theory to classify all affine symmetry groups of vertex-transitive polytopes, thereby answering a question of Babai (1977).

In this review article we discuss four recent methods for computing Maurer-Cartan structure equations of symmetry groups of differential equations. Examples include solution of the contact equivalence problem for linear hyperbolic equations and finding a contact transformation between the generalized Hunter-Saxton equation and the Euler-Poisson equation.

This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups all in analogy with superlines, superplanes etc. The main idea is that the bose-fermi $\pm1$ statistics between Grassmannn coordinates is now replaced by a general braid statistics $\Psi$, typically given by a Yang-Baxter matrix $R$. Most of the algebraic proofs are best done by drawing knot and tangle diagrams, yet most constructions in supersymmetry appear to generalise well. Particles of braid statistics exist and can be expected to be described in this way. At the same time, we find many applications to ordinary quantum group theory: how to make quantum-group covariant (braided) tensor products and spin chains, action-angle variables for quantum groups, vector addition on $q$-Minkowski space and a semidirect product q-Poincar\'e group are among the main applications so far. Every quantum group can be viewed as a braided group, so the theory contains quantum group theory as well as supersymmetry. There also appears to be a rich theory of braided geometry, more general than super-geometry and including aspects of quantum geometry. Braided-derivations obey a braided-Leibniz rule and recover the usual Jackson $q$-derivative as the 1-dimensional case.

This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups all in analogy with superlines, superplanes etc. The main idea is that the bose-fermi $\pm1$ statistics between Grassmannn coordinates is now replaced by a general braid statistics $\Psi$, typically given by a Yang-Baxter matrix $R$. Most of the algebraic proofs are best done by drawing knot and tangle diagrams, yet most constructions in supersymmetry appear to generalise well. Particles of braid statistics exist and can be expected to be described in this way. At the same time, we find many applications to ordinary quantum group theory: how to make quantum-group covariant (braided) tensor products and spin chains, action-angle variables for quantum groups, vector addition on $q$-Minkowski space and a semidirect product q-Poincar\'e group are among the main applications so far. Every quantum group can be viewed as a braided group, so the theory contains quantum group theory as well as supersymmetry. There also appears to be a rich theory of braided geometry, more general than super-geometry and including aspects of quantum geometry. Braided-derivations obey a braided-Leibniz rule and recover the usual Jackson $q$-derivative as the 1-dimensional case.

This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups all in analogy with superlines, superplanes etc. The main idea is that the bose-fermi $\pm1$ statistics between Grassmannn coordinates is now replaced by a general braid statistics $\Psi$, typically given by a Yang-Baxter matrix $R$. Most of the algebraic proofs are best done by drawing knot and tangle diagrams, yet most constructions in supersymmetry appear to generalise well. Particles of braid statistics exist and can be expected to be described in this way. At the same time, we find many applications to ordinary quantum group theory: how to make quantum-group covariant (braided) tensor products and spin chains, action-angle variables for quantum groups, vector addition on $q$-Minkowski space and a semidirect product q-Poincar\'e group are among the main applications so far. Every quantum group can be viewed as a braided group, so the theory contains quantum group theory as well as supersymmetry. There also appears to be a rich theory of braided geometry, more general than super-geometry and including aspects of quantum geometry. Braided-derivations obey a braided-Leibniz rule and recover the usual Jackson $q$-derivative as the 1-dimensional case.

Bell System Technical Journal, 51: 2. February 1972 pp 507-542. Controllability and Observability in Linear Time-Variable Networks With Arbitrary Symmetry Groups. (Rubin, H.; Meadows, H.E.)

For a rotating dust with a 3-dimensional symmetry group all possible metric forms can be classified and, within each class, explicitly written out. This is made possible by the formalism of Pleba\'nski based on the Darboux theorem. In the resulting coordinates, the Killing vector fields (if any exist) assume a special form. Each Killing vector field may be either spanned on the fields of velocity and rotation or linearly independent of them. By considering all such cases one arrives at the classification. With respect to the structures of the groups, this is just the Bianchi classification, but with all possible orientations of the orbits taken into account. In this paper, which is part 1 of a 3-part series, all solutions are considered for which two Killing fields are spanned on velocity and rotation. The solutions of Lanczos and G\"{o}del are identified as special cases, and their new invariant definitions are provided. In addition, a new invariant definition is given of the Ozsvath class III solution.

Symmetries play a crucial role in electroweak symmetry breaking models with non-minimal Higgs content. Within each class of these models, it is desirable to know which symmetry groups can be implemented via the scalar sector. In N-Higgs-doublet models, this classification problem was solved only for N=2 doublets. Very recently, we suggested a method to classify all realizable finite symmetry groups of Higgs-family transformations in the three-Higgs-doublet model (3HDM). Here, we present this classification in all detail together with an introduction to the theory of solvable groups, which play the key role in our derivation. We also consider generalized-CP symmetries, and discuss the interplay between Higgs-family symmetries and CP-conservation. In particular, we prove that presence of the Z_4 symmetry guarantees the explicit CP-conservation of the potential. This work completes classification of finite reparametrization symmetry groups in 3HDM.

For an almost complex structure $J$ in dimension 6 with nondegenerate Nijenhuis tensor $N_J$, the automorphism group $G=Aut(J)$ of maximal dimension is the exceptional Lie group $G_2$. In this paper we establish that the sub-maximal dimension of automorphism groups of almost complex structures with nondegenerate $N_J$, i.e. the largest realizable dimension that is less than 14, is $\dim G=10$. Next we prove that only 3 spaces realize this, and all of them are strictly nearly (pseudo-) K\"ahler and globally homogeneous. Moreover, we show that all examples with $\dim Aut(J)=9$ have semi-simple isotropy.

We define and show the existence of the quantum symmetry group of a Hilbert module equipped with an orthogonal filtration. Our construction unifies the constructions of Banica-Skalski's quantum symmetry group of a C*-algebra equipped with an orthogonal filtration and Goswami's quantum isometry group of an admissible spectral triple.

A connection between creation of toroidal moments and breaking of the relativistic crystalline group associated to a given crystal, is presented in this paper. Indeed, if magnetoelectric effects exist, the interaction between electrons and elementary magnetic cells appears in such a way that the resulting local polarization and magnetization break the local relativistic crystalline symmetry. Therefore, Goldstone bosons, also associated to toroidal moments, are created and, as a consequence, corresponding toroidal phases in crystals. The list of the Shubnikov groups compatible with this kind of phases is given and possible consequences in superconductor theory in magnetoelectric crystals are examined.

Modified Klein-Gordon-Fock equations were obtained on the basis of one-dimensional chaotic dynamics. The original Lagrangians were found. The concepts of \textit{m}-exponential map and groups with broken symmetry are introduced. A system of bitrial orthogonal functions is considered.

Operator fractional Brownian motions (OFBMs) are zero mean, operator self-similar (o.s.s.), Gaussian processes with stationary increments. They generalize univariate fractional Brownian motions to the multivariate context. It is well-known that the so-called symmetry group of an o.s.s. process is conjugate to subgroups of the orthogonal group. Moreover, by a celebrated result of Hudson and Mason, the set of all exponents of an operator self-similar process can be related to the tangent space of its symmetry group. In this paper, we revisit and study both the symmetry groups and exponent sets for the class of OFBMs based on their spectral domain integral representations. A general description of the symmetry groups of OFBMs in terms of subsets of centralizers of the spectral domain parameters is provided. OFBMs with symmetry groups of maximal and minimal types are studied in any dimension. In particular, it is shown that OFBMs have minimal symmetry groups (as thus, unique exponents) in general, in the topological sense. Finer classification results of OFBMs, based on the explicit construction of their symmetry groups, are given in the lower dimensions 2 and 3. It is also shown that the parametrization of spectral domain integral representations are, in a suitable sense, not affected by the multiplicity of exponents, whereas the same is not true for time domain integral representations.

Operator fractional Brownian motions (OFBMs) are zero mean, operator self-similar (o.s.s.), Gaussian processes with stationary increments. They generalize univariate fractional Brownian motions to the multivariate context. It is well-known that the so-called symmetry group of an o.s.s. process is conjugate to subgroups of the orthogonal group. Moreover, by a celebrated result of Hudson and Mason, the set of all exponents of an operator self-similar process can be related to the tangent space of its symmetry group. In this paper, we revisit and study both the symmetry groups and exponent sets for the class of OFBMs based on their spectral domain integral representations. A general description of the symmetry groups of OFBMs in terms of subsets of centralizers of the spectral domain parameters is provided. OFBMs with symmetry groups of maximal and minimal types are studied in any dimension. In particular, it is shown that OFBMs have minimal symmetry groups (as thus, unique exponents) in general, in the topological sense. Finer classification results of OFBMs, based on the explicit construction of their symmetry groups, are given in the lower dimensions 2 and 3. It is also shown that the parametrization of spectral domain integral representations are, in a suitable sense, not affected by the multiplicity of exponents, whereas the same is not true for time domain integral representations.

We investigate more generally the possible unification Yang-Mills groups $G_{YM}$ and representations with CP as a gauge symmetry. Besides the possible Yang-Mills groups $E_8$, $E_7$, $SO(2n+1)$, $SO(4n)$, $SP(2n)$, $G_2$ or $F_4$ (or a product of them) which only allow self-contragredient representations, we present other unification groups $G_{YM}$ and representations which may allow CP as a gauge symmetry. These include especially $SU(N)$ containing Weyl fermions and their CP-conjugates from low-energy spectra in a basic irreducible representation (IR). Such an example is the 496-dimensional basic IR (on antisymmetric tensors of rank two) of $SU(32)$ containing $SO(32)$ as a subgroup in the adjoint IR, or $SU(248)$ in a fundamental IR containing $E_8$ as a subgroup in the adjoint IR. Our consideration also leads to the construction of a physical operator (CP) intrincially as an inner automorphism of order higher than two for the unification group. We have also generalized the possible groups as unification $G_{YM}$ to include non-semisimple Lie groups with CP arising as a gauge symmetry. In this case with $U(1)$ ideals in the $G_{YM}$, we found that the $U_Y(1)$ for weak hypercharge in the standard model or a $U(1)$ gauge symmetry at low energies in general is traceless. Possible relevance to superstring theory is also briefly discussed. We expect that our results may open new alternatives for unified model building, especially with deeper or more generalized understanding of anomaly-free theories.

The possible external couplings of an extended non-relativistic classical system are characterized by gauging its maximal dynamical symmetry group at the center-of-mass. The Galilean one-time and two-times harmonic oscillators are exploited as models. The following remarkable results are then obtained: 1) a peculiar form of interaction of the system as a whole with the external gauge fields; 2) a modification of the dynamical part of the symmetry transformations, which is needed to take into account the alteration of the dynamics itself, induced by the {\it gauge} fields. In particular, the Yang-Mills fields associated to the internal rotations have the effect of modifying the time derivative of the internal variables in a scheme of minimal coupling (introduction of an internal covariant derivative); 3) given their dynamical effect, the Yang-Mills fields associated to the internal rotations apparently define a sort of Galilean spin connection, while the Yang-Mills fields associated to the quadrupole momentum and to the internal energy have the effect of introducing a sort of dynamically induced internal metric in the relative space.

We present some of the group theoretic properties of reversing symmetry groups, and classify their structure in simple cases that occur frequently in several well-known groups of dynamical systems.

I re-derive the lowest order effective Lagrangian for electro-weak symmetry breaking without the use of Goldstone's theorem for spontaneously broken global symmetries and without the assumption of a custodial symmetry. I consider the breaking of a local symmetry with gauge group G down to an Abelian sub-group K and construct a gauge-invariant functional with one free parameter v, such that v=0 corresponds to a gauge theory with gauge group G, while v --> infinity corresponds to a gauge theory with gauge group K.

We consider irreducible unitary representations $A_i$ of G=SO(n+1,1) with the same infinitesimal character as the trivial representation and representations $B_j$ of H=SO(n,1) with the same properties and discuss H-equivariant homomorphisms Hom_H($A_i,B_j$). For tempered representations our results confirm the predictions of conjectures by B. Gross and D. Prasad.

We consider irreducible unitary representations $A_i$ of G=SO(n+1,1) with the same infinitesimal character as the trivial representation and representations $B_j$ of H=SO(n,1) with the same properties and discuss H-equivariant homomorphisms Hom_H($A_i,B_j$). For tempered representations our results confirm the predictions of conjectures by B. Gross and D. Prasad.

For each $n\leq 6$, we characterize all the groups which can occur as either the orientation preserving topological symmetry group or the topological symmetry group of some embedding of $K_n$ in $S^3$.

We consider the following class of unitary representations $\pi $ of some (real) Lie group $G$ which has a matched pair of symmetries described as follows: (i) Suppose $G$ has a period-2 automorphism $\tau $, and that the Hilbert space $\mathbf{H} (\pi)$ carries a unitary operator $J$ such that $J\pi =(\pi \circ \tau)J$ (i.e., selfsimilarity). (ii) An added symmetry is implied if $\mathbf{H} (\pi)$ further contains a closed subspace $\mathbf{K}_0 $ having a certain order-covariance property, and satisfying the $\mathbf{K}_0 $-restricted positivity: $ < v \mid Jv > \ge 0$, $\forall v\in \mathbf{K}_0 $, where $ < \cdot \mid \cdot >$ is the inner product in $\mathbf{H} (\pi)$. From (i)--(ii), we get an induced dual representation of an associated dual group $G^c$. All three properties, selfsimilarity, order-covariance, and positivity, are satisfied in a natural context when $G$ is semisimple and hermitean; but when $G$ is the $(ax+b)$-group, or the Heisenberg group, positivity is incompatible with the other two axioms for the infinite-dimensional irreducible representations. We describe a class of $G$, containing the latter two, which admits a classification of the possible spaces $\mathbf{K}_0 \subset \mathbf{H} (\pi)$ satisfying the axioms of selfsimilarity and order-covariance.

Analytical solutions of variable coefficient nonlinear Schr\"odinger equations having four-dimensional symmetry groups which are in fact the next closest to the integrable ones occurring only when the Lie symmetry group is five-dimensional are obtained using two different tools. The first tool is to use one dimensional subgroups of the full symmetry group to generate solutions from those of the reduced ODEs (Ordinary Differential Equations), namely group invariant solutions. The other is by truncation in their Painlev\'e expansions.

We present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a semisimple Lie group $G$ and a semisimple subgroup $G'$, and between their composition factors. Our method decribes the restriction of these operators to the $K'$-isotypic components, $K'\subseteq G'$ a maximal compact subgroup, and reduces the representation theoretic problem to an infinite system of scalar equations of a combinatorial nature. For rank one orthogonal and unitary groups and spherical principal series representations we calculate these relations explicitly and use them to classify intertwining operators. We further show that in these cases automatic continuity holds, i.e. every intertwiner between the Harish-Chandra modules extends to an intertwiner between the Casselman--Wallach completions, verifying a conjecture by Kobayashi. Altogether, this establishes the compact picture of the recently studied symmetry breaking operators for orthogonal groups by Kobayashi--Speh, gives new proofs of their main results and extends them to unitary groups.

We present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a semisimple Lie group $G$ and a semisimple subgroup $G'$, and between their composition factors. Our method decribes the restriction of these operators to the $K'$-isotypic components, $K'\subseteq G'$ a maximal compact subgroup, and reduces the representation theoretic problem to an infinite system of scalar equations of a combinatorial nature. For rank one orthogonal and unitary groups and spherical principal series representations we calculate these relations explicitly and use them to classify intertwining operators. We further show that in these cases automatic continuity holds, i.e. every intertwiner between the Harish-Chandra modules extends to an intertwiner between the Casselman--Wallach completions, verifying a conjecture by Kobayashi. Altogether, this establishes the compact picture of the recently studied symmetry breaking operators for orthogonal groups by Kobayashi--Speh, gives new proofs of their main results and extends them to unitary groups.