Isotopic Liftings of Clifford Algebras and Applications in Elementary Particle Mass Matrices

Isotopic liftings of algebraic structures are investigated in the context of Clifford algebras, where it is defined a new product involving an arbitrary, but fixed, element of the Clifford algebra. This element acts as the unit with respect to the introduced product, and is called isounit. We construct isotopies in both associative and non-associative arbitrary algebras, and examples of these constructions are exhibited using Clifford algebras, which although associative, can generate the octonionic, non-associative, algebra. The whole formalism is developed in a Clifford algebraic arena, giving also the necessary pre-requisites to introduce isotopies of the exterior algebra. The flavor hadronic symmetry of the six u,d,s,c,b,t quarks is shown to be exact, when the generators of the isotopic Lie algebra are constructed, and the unit of the isotopic Clifford algebra is shown to be a function of the six quark masses. The limits constraining the parameters, that are entries of the representation of the isounit in the isotopic group SU(6), are based on the most recent limits imposed on quark masses.     

Quantization and representation theory of finiteW algebras

In this paper we study the finitely generated algebras underlyingW algebras. These so called finiteW algebras are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings ofsl ₂ into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finiteW algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finiteW symmetry. In the second part we BRST quantize the finiteW algebras. The BRST cohomology is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finiteW algebras in one stroke. Examples are given. In the last part of the paper we study the representation theory of finiteW algebras. It is shown, using a quantum version of the generalized Miura transformation, that the representations of finiteW algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finiteW algebras.     

Finite-dimensional representations of the euclidean group in the plane

In this article two theorems are given which permit, together with the concept of a representation vector diagram, to classify all (linear) finite-dimensional representations of the algebra and group E ₂ which are induced by a master representation on the place of the universal enveloping algebra of the algebra E ₂. Apart from a classification of the finite-dimensional representations, the two theorems make it possible to obtain the matrix elements of these representations for both, algebra and group, in explicit form. The material contained in this letter forms part of an analysis of indecomposable (finite- and infinite-dimensional) representations of the algebra and group E ₂ which is contained in Reference [1]. No proofs will be given in this letter. We refer instead to [1].     

On the relation between types of local algebras in different global representations

LetA,A(O), ^d, be a theory of local observables. We show that there are relations between the Connes-von Neumann types of the algebras belonging to a different global representation. For example if one representation is the vacuum representation such that the wedge algebra is of type III₁ then this also is the case for other representations, provided these are connected with the vacuum by large translations.     

First order deformations of Lie algebra representations,E (3) and Poincaré examples

A classification of first order deformations of Lie algebra representations by the use of a cohomology group is studied. A method is proposed for calculating this group for the case of algebras which are semi-direct products. The role of unitarity of the representations is exhibited. Applications are made for the Poincaré andE(3) algebras.     

The relation between quantumW algebras and Lie algebras

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrarysl ₂ embeddings we show that a large set of quantumW algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set contains many knownW algebras such asW _N andW ₃ ⁽²⁾ . Our formalism yields a completely algorithmic method for calculating theW algebra generators and their operator product expansions, replacing the cumbersome construction ofW algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that anyW algebra in can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Thereforeany realization of this semisimple affine Lie algebra leads to a realization of theW algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolusions for all algebras in. Some examples are explicitly worked out.     

Nonlinear deformed su(2) algebras involving two deforming functions

The most common nonlinear deformations of the su(2) Lie algebra, introduced by Polychronakos and Roek, involve a single arbitrary function ofJ _o and include the quantum algebra su _q (2) as a special case. In the present contribution, less common nonlinear deformations of su(2), introduced by Delbecq and Quesne and involving two deforming functions ofJ _o, are reviewed. Such algebras include Witten's quadratic deformation of su(2) as a special case. Contrary to the former deformations, for which the spectrum ofJo is linear as for su(2), the latter give rise to exponential spectra, a property that has aroused much interest in connection with some physical problems. Another interesting algebra of this type, denoted byA _q ⁺ (1), has two series of (N+1)-dimensional unitary irreducible representations, whereN=0, 1, 2, ... To allow the coupling of any two such representations, a generalization of the standard Hopf axioms is proposed. The resulting algebraic structure, referred to as a two-colour quasitriangular Hopf algebra, is described.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.Directeur de recherches FNRS.     

Some remarks on the algebra of Eddington'sE numbers

This paper reviews the algebra of Eddington'sE numbers and identifies those points where Eddington anticipated results of current interest. He discovered the Majorana spinors, and was responsible for the standard ⁵ notation as well as the notion of chirality. Furthermore, Eddington defined Clifford algebras in eight and nine dimensions which are now appearing in grand unified gauge and supersymmetric theories. A point which Eddington cleared up, yet is still misunderstood, is that the Dirac algebra corresponds to afive-dimensional base space.     

Normal product states for fermions and twisted duality for CCR- and CAR-type algebras with application to the Yukawa2 quantum field model

We present sufficient conditions that imply duality for the algebras of local observables in all Abelian sectors of all locally normal, irreducible representations of a field algebra if twisted duality obtains in one of these representations. It is verified that the Yukawa₂ model satisfies these conditions, yielding the first proof of duality for the observable algebra in all coherent charge sectors in this model. This paper also constitutes the first verification of the assumptions of the axiomatic study of the structure of superselection sectors by Doplicher, Haag and Roberts in an interacting model with nontrivial sectors. The existence of normal product states for the free Fermi field algebra and, thus, the verification of the funnel property for the associated net of local algebras are demonstrated.     

Highest weight representations of the Neveu-Schwarz and Ramond algebras

We construct a family of representationsK ^,w of the Neveu-Schwarz and Ramond algebras, which generalize the Fock representations of the Virasoro algebra. We show that the representationsK ^,w are intertwined by a vertex operator.The above results are used to give the proof of the conjectured formulas for the determinant of the contravariant form on the highest weight representations of the Neveu-Schwarz and Ramond algebras. Further results on the representation theory of the latter are derived from the determinant formulas.Partially supported by the National Science Foundation through the Mathematical Sciences Research InstitutePartially supported by NSF grant MCS-8201260     

Representations of the cyclically symmetric q-deformed algebra U q(so3)

An algebra homomorphism from the nonstandard q-deformed (cyclically symmetric) algebra U _q(so₃) to the extension Û _q(sl₂) of the Hopf algebra U _q(sl₂) is constructed. Not all irreducible representations (IR) of U _q(sl₂) can be extended to representations of Û _q(sl₂). Composing the homomorphism with irreducible representations of Û _q(sl₂) we obtain representations of U _q(so₃). Not all of these representations of U _q(so₃) are irreducible. Reducible representations of U _q(so₃) are decomposed into irreducible components. In this way we obtain all IR of U _q(so₃) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so₃ when q 1.     

Linear operators in Clifford algebras

We consider the real vector space structure of the algebra of linear endomorphisms of a finite-dimensional real Clifford algebra (2, 4, 5, 6, 7, 8). A basis of that space is constructed in terms of the operators M _{eI, eJ} defined by xe _I · x · e _J , where the e _I are the generators of the Clifford algebra and I is a multi-index (3, 7).In particular, it is shown that the family (M _{eI, eJ} ) is exactly a basis in the even case.     

Idempotents of Clifford Algebras

A classification of idempotents of Clifford algebras C ^p,q is presented. It is shown that using isomorphisms between Clifford algebras C ^p,q and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one-sided ideals in Clifford algebras. Some low-dimensional examples are discussed.     

Direct product and decomposition of certain physically important algebras

I consider the direct product algebra formed from two isomorphic Clifford algebras. More specifically, for an element x in each of the two component algebras I consider elements in the direct product space with the form x x. I show how this construction can be used to model the algebraic structure of particular vector spaces with metric, to describe the relationship between wavefunction and observable in examples from quantum mechanics, and to express the relationship between the electromagnetic field tensor and the stress-energy tensor in electromagnetism. To enable this analysis I introduce a particular decomposition of the direct product algebra.     

Extended super-Kač-Moody algebras and their super-derivation algebras

We study theN-extended super-Ka-Moody algebras, i.e. extensions of the Lie algebra of the loop group over the super-circleA _N . The extensions are characterized by 2-cocycles which are computed in terms of the cyclic cohomology of the Grassmann algebra withN generators. The graded algebra of super-derivations compatible with each extension is determined. The casesN=1,2,3 are examined in detail and their relation with the Ademollo et al. superconformal algebras is discussed. We examine the possibility of defining new superconformal algebras which, forN>1, generalize theN=1 Ramond-Neveu-Schwarz algebra.     

Highest weight representations of quantum current algebras

We study the highest weight and continuous tensor product representations ofq-deformed Lie algebras through the mappings of a manifold into a locally compact group. As an example the highest weight representation of theq-deformed algebra sl_q(2,) is calculated in detail.Alexander von Humboldt-Stiftung fellow. On leave from Institute of Physics, Chinese Academy of Sciences, Beijing, P.R. China.     

Cλ-Extended Oscillator Algebras and Some of Their Deformations and Applications to Quantum Mechanics

C-extended oscillator algebras generalizing the Calogero—Vasiliev algebra,where C is the cyclic group of order , are studied both from mathematical andapplied viewpoints. Casimir operators of the algebras are obtained and used toprovide a complete classification of their unitary irreducible representations underthe assumption that the number operator spectrum is nondegenerate. Deformedalgebras admitting Casimir operators analogous to those of their undeformedcounterparts are looked for, yielding three new algebraic structures. One of themincludes the Brzezi´nski et al. deformation of the Calogero—Vasiliev algebra as aspecial case. In its bosonic Fock-space representation, the realization ofC-extended oscillator algebras as generalized deformed oscillator ones is shown toprovide a bosonization of several variants of supersymmetric quantum mechanics:parasupersymmetric quantum mechanics of order p = – 1 for any , as wellas pseudosupersymmetric and orthosupersymmetric quantum mechanics of ordertwo for = 3.     

W-Algebras,new rational models and completeness of thec=1 classification

Two series ofW with two generators are constructed from chiral vertex operators of a free field representation. Ifc=1–24k, there exists aW(2, 3k) algebra for k ₊/2 and aW(2, 8k) algebra for k ₊/4. All possible lowest-weight representations, their characters and fusion rules are calculated proving that these theories are rational. It is shown, that these non-unitary theories complete the classification of all rational theories with effective central chargec _eff=1. The results are generalized to the case of extended supersymmetric conformal algebras.