Non-linear realization of superconformal symmetry

Within a framework of the non-linear realization, we investigate the transformation law and the effective lagrangian when superconformal symmetry is spontaneously broken down to super-Poincaré symmetry. We show that the determinant of the “supervierbein” provides an effective lagrangian which describes the interaction among the dilaton, the “axion” and the dilatino. We also briefly discuss the case in which the superconformal group is broken down to the Poincaré group.     

Quonic realization of the deformed Heisenberg-Weyl algebra

A consistent realization of an operatorial deformed Heisenberg-Weyl (HW) algebra is given. The explicit construction of the deformation matrix is discussed. The corresponding generalized Fock-like space is analyzed.     

Time-dependent realization of the infinite-dimensional hydrogen algebra

We use the Hamiltonian formalism to investigate the Katzin–Levine model of a time-dependent Kepler problem. This formalism enables us to define Lie products in terms of Poisson brackets and obtain a time-dependent realization of centerless twisted (or standard) Kac–Moody algebras of so(N+1). We also show that the classical solutions of the model are modulated conic sections and derive a generalized Kepler equation for the time dependence.     

Non-Linear realization of supersymmetry algebra from supersymmetric constraint

We discuss spontaneous symmetry breaking of global supersymmetry for a single scalar superfield in an arbitrary Kähler manifold. We show that when the curvature of the manifold goes to infinity (or, equivalently, the masses of the scalar partners of the goldstino go to infinity) a non-linear realization of supersymmetry is obtained. The model can be described, in perfect analogy to the ordinary σ-models, by means of a supersymmetric constraint on the superfield Φ, of the form Φ²=0. The non-linear realization we obtain is different from that of Volkov and Akulov. The differences among the two realizations are discussed.     

Two-field coset space realization of the spin-4 current algebra

A two-field realization of the spin-4 current algebra is obtained through the coset space approach. The transformation properties of these two fields are also deduced through the composition law, which turns out to be a simple generalization of the corresponding diffeomorphic transformation.     

Dynamical realization of l-conformal Galilei algebra and oscillators

The method of nonlinear realizations is applied to the l-conformal Galilei algebra to construct a dynamical system without higher derivative terms in the equations of motion. A configuration space of the model involves coordinates, which parametrize particles in d spatial dimensions, and a conformal mode, which gives rise to an effective external field. It is shown that trajectories of the system can be mapped into those of a set of decoupled oscillators in d dimensions.     

A novel realization of the Virasoro algebra in number state space

The group of diffeomorphisms is crucial in quantum computing. Representing it by vector fields over a d-manifold, d?2, accounting for both projective action and conformal symmetry at the quantum mechanical level, requires the direct-sum decomposition of tensor product for non-compact algebras, viable only for su(1,1). As a step towards the solution, a realization of the (d=1) Virasoro algebra Vir∼Diff₊(S(1)) in the universal envelope of su(1,1) (and h(1)) is presented, which is simple in the discrete positive series irreducible unitary representation of su(1,1).     

Superspace geometrical realization of the N-extended super Virasoro algebra and its dual

We derive properties of N-extended super Virasoro algebras. These include adding central extensions, identification of all primary fields and the action of the adjoint representation on its dual. The final result suggest identification with the spectrum of fields in supergravity theories and superstring/M-theory constructed from NSR N-extended supersymmetric Virasoro algebras.     

Some remarks on BRS transformations,anomalies and the cohomology of the Lie algebra of the group of gauge transformations

We show that ghosts in gauge theories can be interpreted as Maurer-Cartan forms in the infinite dimensional group ? of gauge transformations. We examine the cohomology of the Lie algebra of ? and identify the coboundary operator with the BRS operator. We describe the anomalous terms encountered in the renormalization of gauge theories (triangle anomalies) as elements of these cohomology groups.     

The blob algebra and the periodic Temperley-Lieb algebra

We determine the structure of two variations on the Temperley-Lieb algebra, both used for dealing with special kinds of boundary conditions in statistical mechanics models. The first is a new algebra, the blob algebra. We determine both the generic and all the exceptional structures for this two parameter algebra. The second is the periodic Temperley-Lieb algebra. The generic structure and part of the exceptional structure of this algebra have already been studied. We complete the analysis using results from the study of the blob algebra.     

Irreducible Lie algebra extensions of the Poincaré algebra

We use cohomology of Lie algebras to analyse the abelian extensions of the Poincaré algebraP. We study particularly the irreducible and truly irreducible extensions: some irreducibility criteria are proved and applied to obtain a classification of types of irreducible abelian extensions ofP. We give a characterization of the minimal essential extensions in terms of truly irreducible extensions.     

Irreducible Lie algebra extensions of the Poincaré algebra

We analyse the extensions of the Poincaré algebraP with arbitrary kernels. The main tool is a reduction theorem which generalizes the Hochschild-Serre theorem forn=2. This reduction theorem is proved and used to investigate the structure of the Lie algebras obtained by extension.We look particularly for the irreducible and -irreducible extensions ofP and we classify the types of irreducible extensions with arbitrary kernels.     

Free-field realization of theWKP(N) algebra

TheW _KP ^(N) algebra has been identified with the second Hamiltonian structure in theNth Hamiltonian pair of the KP hierarchy. In this Letter, by constructing the Miura map that decomposes the second Hamiltonian structure in theNth pair of the KP hierarchy, we show thatW _KP ^(N) can also be decomposed toN independent copies ofW _KP ⁽¹⁾ algebras, therefore its free-field realization can be worked out by constructing free fields for each copy ofW _KP ⁽¹⁾ . In this way, the free fields may consist ofN + 2n number of bosons, among them, 2n are in pairs, wheren is an arbitrary integer between 1 andN. We also express the currents ofW _KP ^(N) in terms of the currents ofN —n copies of U(1) andn copies of SL(2,R) _k algebras with levelk = 1. By reductions, we give similar results forW ^(N) andW ₃ ⁽²⁾ algebra.     

New 3-mode bosonic operator realization of SU（2） Lie algebra： From the point of view of squeezing

We consider the quantum mechanical SU(2) transformation e2λ JzJ± e-2λJz= e±2λJ± as if the meaning of squeezing with e±2λbeing squeezing parameter. By studying SU(2) operators(J±,Jz) from the point of view of squeezing we find that(J±,Jz) can also be realized in terms of 3-mode bosonic operators. Employing this realization, we find the natural representation(the eigenvectors of J+ or J-) of the 3-mode squeezing operator e2λ Jz. The idea of considering quantum SU(2) transformation as if squeezing is liable for us to obtain the new bosonic operator realization of SU(2) and new squeezing operators.     

Meanders and the Temperley-Lieb algebra

The statistics of meanders is studied in connection with the Temperley-Lieb algebra. Each (multi-component) meander corresponds to a pair of reduced elements of the algebra. The assignment of a weightq per connected component of meander translates into a bilinear form on the algebra, with a Gram matrix encoding the fine structure of meander numbers. Here, we calculate the associated Gram determinant as a function ofq, and make use of the orthogonalization process to derive alternative expressions for meander numbers as sums over correlated random walks.     

The spectrum of the algebra of skew differential operators and the irreducible representations of the quantum Heisenberg algebra

The left spectrum of a wide class of the algebras of skew differential operators is described. As a sequence, we determine and classify all the algebraically irreducible representations of the quantum Heisenberg algebra over an arbitrary field.     

Exchange algebra and the Drinfeld-Sokolov theorem

We analyse the relation between the exchange algebra and the separation of the chiralities in classical Toda field theory. We show that there exists a conformally covariant Bloch wave basis such that the two chiralities commute. In terms of this basis we then reconstruct the periodic and local solution of Toda field theory.Work supported by Fondazione Angelo Della Riccia     

q-bosons and the Lie-deformed Heisenberg algebra

It is shown that the non-Hermitian realization of a Lie-deformed Heisenberg algebra given by Jannussis et al. is closely related with the q-Heisenberg-Weyl algebra of Biedenharn and Macfarlane with q being a phase (q = e^iθ, with θ real). The physical implications of this result are stressed.     

Wakimoto realization of drinfeld current for the elliptic quantum algebra Uq,p( $$ \widehat{sl_3 } $$ )

We study a free field realization of the elliptic quantum algebra U_q,p($ \widehat{sl_3 } $ \widehat{sl_3 } ) for arbitrary level k. We give the free field realization of elliptic analog of Drinfeld current associated with U_q,p($ \widehat{sl_3 } $ \widehat{sl_3 } ) for arbitrary level k. In the limit p → 0, q → 1 our realization reproduces Wakimoto realization for the affine Lie algebra $ \widehat{sl_3 } $ \widehat{sl_3 } .