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.     

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.     

The BRS algebra of a free minimal differential algebra

We construct the Weil and the universal BRS algebras of theories that can have as a gauge symmetry a free minimal differential (Sullivan) algebra, the natural extension of Lie algebras allowing the definition of p-form gauge potentials (p > 1). The geometrical meaning of these p-form gauge potentials can be understood with the notion of a Quillen superconnection.     

Crystal algebra

We define the crystal algebra, an algebra which has a base of elements of crystal bases of a quantum group. The multiplication is defined by the tensor product rule of crystal bases. A universal n-colored crystal algebra is defined. We study the relation between those algebras and the tensor algebras of the crystal algebra of U _q (sl(2)) and give a presentation by generators and relations for the case of U _q (sl(n)).     

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.     

Yangian construction of the Virasoro algebra

We show that a Yangian construction based on the algebra of an infinite number of harmonic oscillators (i.e. a vibrating string) terminates after one step, yielding the Virasoro algera.     

Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra

Let be a complex simple Lie algebra. We show that whent is not a root of 1 all finite dimensional representations of the quantum analogU _t are completely reducible, and we classify the irreducible ones in terms of highest weights. In particular, they can be seen as deformations of the representations of the (classical)U .     

States on the current algebra

We introduce the field algebra Σ$\text{D}$(M;ⁿ?ⁿ$\text{g}$) associated with the current algebra $\text{D}$_r(M;$\text{g}$) for the Lie algebra $\text{g}$ over physical space M. The Heisenberg magnet model is generalized to this continuum. It is shown that the Hamiltonian can be given meaning as implementing a derivation of the field algebra in certain representations.We introduce new representations of the current algebra. For example, if G = SU(2), a representation in L²(R³)?³ is [σ(?)F]^j = ε^jkl?_kψ_l for (?_k) = ? in $\text{D}$_r(M;$\text{g}$)(ψ_l = F. This has cyclic subrepresentations with prime parts.     

Spinor algebra of the one-component lattice fermions

We develop a general formalism to translate the Susskind one-component theory of free fermions in a formulation with conventional Dirac spinors. It results that Susskind theory has parity-violating and flavour-changing terms which vanish only in the continuum limit. Gauge fields destroy the equivalence between the one-component and the conventional formulations. In particular the one-component fermions respond to a gauge field as they were in a curved space-time. However, this gravity effect vanishes if one takes a naive continuum limit.     

The three-dimensional origin of the classifying algebra

It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra.     

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.     

Scattering in the CAR algebra

We obtain some results on time evolution in quasi local algebras, which are used to derive the scattering of quasi free evolution groups in the CAR algebra by certain inner perturbations.     

Characterizations of the quantum Witt algebra

The q-analogues of some concepts in the theory of nonassociative algebras are introduced and two characterizations are given for the quantum Witt algebra.     

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.     

Singular vectors of the Neveu-Schwarz algebra

We give an explicit expression for some singular vectors of highest weight representations of the Neveu-Schwarz algebra.     

A diagrammatic categorification of the fermion algebra

In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional （1D） fermion algebra, and we investigate the properties of this category. The categorical analogues of the Fock states are some kind of 1-morphisms in our category, and the dimension of the vector space of 2-morphisms is exactly the inner product of the corresponding Fock states. All the results in our categorical framework coincide exnetlv with those in normal quantum mechanics.     

Singular vectors of the Virasoro algebra

We present an explicit construction of the singular (or null) vectors in highest weight Verma modules.