High efficiency redundant binary number representations for parallel arithmetic on optical computers

A family of redundant binary number representations, obtained by generalization of the RB (redundant binary) number representation, is introduced. All these number representations are suitable for optical computing and have properties similar to the RB representation. In particular, the p-RB (packed redundant binary) number representation introduced in this work has efficiency greater than both RB and MSD (modified signed digit) representations. With p-RB numbers the algebraic sum is always permitted in constant time for any efficiency value. p-RB representations also fit in a natural way the 2's complement binary number system. Symbolic substitution truth tables for the algebraic sum and several examples of computation are also given.     

On the representations of quantum oscillator algebra

It is shown that for q<1, the quantum oscillator algebra has a supplementary family of representations inequivalent to the usual q-Fock representation, with no counterpart at the limit q=1. They are used to build representations of SU _q (1,1) and E(2) in Schwinger's way.     

Localization of Unitary Braid Group Representations

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e ^πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 R-matrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N > 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science.     

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     

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     

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].     

The cyclic representations of the quantum superalgebra U q osp(2, 1) withq a root of unity

By generalizing De Concini and Kac's cyclic representation theory of quantum groups at roots of unity, the cyclic representations of the quantum superalgebra U _q osp(2, 1) are constructed in three classes: irreducible representations with single multiplicities, irreducible representations with the multiplicities larger than one, and indecomposable representations.This work is supported in part by the National Sciene Foundation in China.     

Signature characters forA 2 andB 2

The signatures of the inner product matrices on a Lei algebra's highest weight representation are encoded in the representation's signature character. We show that the signature characters of a finite-dimensional Lie algebra's highest weight representations obey simple difference equations that have a unique solution once appropriate boundary conditions are imposed. We use these results to derive the signature characters of allA ₂ andB ₂ highest weight representations. Our results extend, and explain, signature patterns analogous to those observed by Friedan, Qiu and Shenker in the Virasoro algebra's representation theory.     

A contribution to group representations in locally convex spaces

Let U be a continuous representation of a (connected) locally compact group G in a separated locally convex space E. It is shown that the study of U is equivalent to the study of a family U _i of continuous representations of G in Fréchet spaces F _i. If U is equicontinuous, the F _i are Banach spaces, and the U _i are realized by isometric operators. When U is topologically irreducible, it is Naïmark equivalent to a Fréchet (or isometric Banach, in the equicontinuous case) continuous representation. Similar results hold for semi-groups.     

Algebraic fermion bosonization

By exploiting the construction of charged field algebras as canonical extensions of CCR current algebras in 1+1 dimensions and nonregular representations of extended algebras, we provide an algebraic construction of local Fermi fields as ultrastrong limits of bosonic variables in all representations which are locally Fock with respect to the ground-state representation of the massless scalar field.     

Trilinear Lorentz invariant forms

Trilinear invariant forms are described over spaces transforming under the so-called elementary representations ofSL(2,C) obtained from the Gel'fand-Naimark principal series by analytic continuation in the representation parameters (among these are all infinite-dimensional completely irreducible representations). All such forms are described using a manifestly covariant technique. The method is based on a natural one-one correspondence between the invariant forms and invariant separately homogeneous distributions (called kernels of the forms) in three complex two-dimensional non-zero vectors; thus the problem is completely reduced to a problem of distribution theory. The kernels display analyticity properties in the representation parameters; the results on this point are only sketched.     

On factor representations and theC*-algebra of canonical commutation relations

A newC*-algebra,A, for canonical commutation relations, both in the case of finite and infinite number of degrees of freedom, is defined. It has the property that to each, not necessarily continuous, representation of CCR there corresponds a representation ofA. The definition ofA is based on the existence and uniqueness of the factor type II₁ representation. Some continuity properties of separable factor representations are proved.     

Induced representations of the two-parameter Jordanian quantum algebra

We construct induced infinite-dimensional representations of the two-parameter quantum algebraUg,h(gl(2)) which is in duality with the deformationGLg,h(2). The representations depend on two representation parameters, but split into one-parameter representations of a one-generator central subalgebra and the three-generator Jordanian quantum subalgebraU (sl(2)), =g + h. The representations of the latter can be mapped to representations in one complex variable, which give anew deformation of the standard one-parameter vector-field realization ofsl(2). These infinite-dimensional representations are reducible for some values of the representation parameters, and then we obtain canonically the finite-dimensional representations ofU (sl(2)). Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. Permanent address of V.K.D.     

Indecomposable representations of the algebra su(2,1)

Representations of the Lie algebra sl(3) with highest weight are analyzed. Invariant subspaces of indecomposable representations are determined. We study the decomposition of these representations with respect to the subalgebras su(2) and su(1,1) (in their obvious imbedding in su(2,1)).For special cases this decomposition gives indecomposable non multiplicity free representations (indecomposable pairs) with highest weight. These were discussed in [1] and appear also in the decomposition so(3,2) su(1,1) of the Rac representation, [7].     

Degenerations and Representations of Twisted Shibukawa–Ueno R-Operators

We study degenerations of the Belavin R-matrices via the infinite dimensional operators defined by Shibukawa–Ueno. We define a two-parameter family of generalizations of the Shibukawa–Ueno R-operators. These operators have finite dimensional representations which include Belavin's R-matrices in the elliptic case, a two-parameter family of twisted affinized Cremmer–Gervais R-matrices in the trigonometric case, and a two-parameter family of twisted (affinized) generalized Jordanian R-matrices in the rational case. We find finite dimensional representations which are compatible with the elliptic to trigonometric and rational degeneration. We further show that certain members of the elliptic family of operators have no finite dimensional representations. These R-operators unify and generalize earlier constructions of Felder and Pasquier, Ding and Hodges, and the authors, and illuminate the extent to which the Cremmer–Gervais R-matrices (and their rational forms) are degenerations of Belavin's R-matrix.     

On the <Emphasis Type="Italic">s</Emphasis>-parameterized Quantization Scheme for Dirac’s Representation theory

Dirac is the founder of quantum mechanical representation theory. By virtue of the technique of integration within an ordered product (IWOP) of operators we introduce s-parameterized form of quantum mechanical coordinate and momentum representations, which are complete. We then point out that s-parameterized representation’s completeness relation is accompanied with operators’ s-ordering, the special cases s=1,0,−1 correspond to normal-ordering, Weyl ordering and antinormal-ordering, respectively. The s-parameterized form of the coherent state representation and the entangled state representation are also derived. In our view, the operators’ s-ordering should be traced back to s-parameterized form of the completeness relation of quantum mechanical coordinate and momentum representations, which is more fundamental. Many operator identities can be derived by virtue of the above mentioned s-parameterized representation’s completeness relations.     

Cyclic monodromy matrices forsl(n) trigonometricR-matrices

The algebra of monodromy matrices forsl(n) trigonometricR-matrix is studied. It is shown that a generic finite-dimensional polynomial irreducible representation of this algebra is equivalent to a tensor product ofL-operators. Cocommutativity of representations is discussed and intertwiners for factorizable representations are written through the Boltzmann weights of thesl(n) chiral Potts model.     

Irreducible highest weight representations of quantum groupsU q (gl(n,C))

Explicit recurrence formulas of canonical realization (boson representation) for quantum enveloping algebrasU _q (gl(n, C)) are given. Using them, irreducible highest weight representations ofU _q (gl(n, C)) are obtained as restriction of representation of Fock space to invariant subspace generated by vacuum as a cyclic vector.     

Nuclear Algebraic Geometry and SO(10)

In this contribution nuclear representations of the Dirac ring, developed over many years, are shown to be a particular case of a theorem in algebraic geometry which at the same time associates them with a Hodge decomposition of a Kaehler manifold. This yields a shape that in some cases is independent of any appeal to a symmetry group. However, because the nuclear representations are in the infinitesimal ring of SO(4) and the internal space of each representation is in a Kaehler (even Calabi-Yau) manifold K; the group SO(10) = SO(4) × K can give additional information. This paper develops the very fruitful symbiosis between algebra and irreducible representations of SO(10) and covers some aspects of string theory.     

On Nested Bethe Ansatz for RTT Algebra of o(2n) Type

We study the highest weight representations of the RTT algebras for the R matrix of o(2n) type by the nested algebraic Bethe ansatz. We show how auxiliary RTT algebra à can be used to find Bethe vectors and Bethe conditions. For special representations, in which representation of RTT algebra à is trivial, the problem was solved by Reshetikhin.