THE ADJOINT REPRESENTATION OF QUANTUM ALGEBRA Uq(sl(2))

Starting from any representation of the Lie algebra on the finite dimensional vector space V we can construct the representation on the space Aut(V). These representations are of the type of ad. That is one of the reasons, why it is important to study the adjoint representation of the Lie algebra on the universal enveloping algebra U(). A similar situation is for the quantum groups U_q(). In this paper, we study the adjoint representation for the simplest quantum algebra U_q(sl(2)) in the case that q is not a root of unity.     

多项式角动量代数的代数表示及实现

本文利用三参数李群求代数表示的方法求出多项式角动量代数的代数表示及其酉表示，找到一个能同时承载李代数及相对应的多项式角动量代数的基底，并在该基底下求出两种代数之间的联系，利用该联系则也可求出多项式角动量代数的代数表示.最后求出多项式角动量代数的单玻色实现及其在有限维多项式函数空间的微分实现. 关键词： 多项式角动量代数 Higgs代数 su(2)代数     

Local Currents for a Deformed Heisenberg–Poincaré Lie Algebra of Quantum Mechanics,and Anyon Statistics

We set out to construct a Lie algebra of local currents whose space integrals, or “charges”, form a subalgebra of the deformed Heisenberg–Poincaré algebra of quantum mechanics discussed by Vilela Mendes, parameterized by a fundamental length scale ℓ. One possible technique is to localize with respect to an abstract single-particle configuration space having one dimension more than the original physical space. Then in the limit ℓ→0, the extra dimension becomes an unobservable, internal degree of freedom. The deformed (1+1)-dimensional theory entails self-adjoint representations of an infinite-dimensional Lie algebra of nonrelativistic, local currents modeled on (2+1)-dimensional space-time. This suggests a new possible interpretation of such representations of the local current algebra, not as describing conventional particles satisfying bosonic, fermionic, or anyonic statistics in two-space, but as describing systems obeying these statistics in a deformed one-dimensional quantum mechanics. In this context, we have an interesting comparison with earlier results of Hansson, Leinaas, and Myrheim on the dimensional reduction of anyon systems. Thus motivated, we introduce irreducible, anyonic representations of the deformed global symmetry algebra. We also compare with the technique of localizing currents with respect to the discrete positional spectrum.     

Infinite conformal symmetry in two-dimensional quantum field theory

We present an investigation of the massless, two-dimentional, interacting field theories. Their basic property is their invariance under an infinite-dimensional group of conformal (analytic) transformations. It is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of Virasoro algebra, and that the correlation functions are built up of the “conformal blocks” which are completely determined by the conformal invariance. Exactly solvable conformal theories associated with the degenerate representations are analyzed. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the systems of linear differential equations.     

Hidden quantum groups inside Kac-Moody algebra

A lattice analogue of the Kac-Moody algebra is constructed. It is shown that the generators of the quantum algebra with the deformation parameterq=exp(iπ/k+h) can be constructed in terms of generators of the lattice Kac-Moody algebra (LKM) with the central chargek. It appears that there exists a natural correspondence between representations of the LKM algebra and the finite dimensional quantum group. The tensor product for representations of the LKM algebra and the finite dimensional quantum algebra is suggested.     

Bethe's Equation Is Incomplete for the XXZ Model at Roots of Unity

We demonstrate for the six vertex and XXZ model parameterized by = –(q+q^-1)/2±1 that when q^2N=1 for integer N2 the Bethe's ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl₂ loop algebra symmetry group of the model. Therefore in this case the Bethe's ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function. We discuss how the evaluation parameters of the finite dimensional representations of the sl₂ loop algebra can be used to complete this specification.     

A Coset-Type Construction for the Deformed Virasoro Algebra

An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra U_q(sl₂). A similar construction is proposed for the elliptic algebra A_q,p(sl₂).     

A mathematical structure for the generalization of conventional algebra

An abstract mathematical framework is presented in this paper as a unification of several deformed or generalized algebra proposed recently in the context of generalized statistical theories intended to treat certain complex thermodynamic or statistical systems. It is shown that, from a mathematical point of view, any bijective function can in principle be used to formulate an algebra in which the conventional algebraic rules are generalized.      

Charged Representations of the Infinite Fermi and Clifford Algebras

The real and quaternionic charge conjugation operators invariant under the infinite-dimensional Clifford algebra, or compatible with the Fermi algebra, are determined. There results a maze of inequivalent irreducible charged representations, all of which are non-Fock. The representation vectors and their charges admit two interpretations besides those of spinors or states of quantum fields: as wavelets on the circle, with charge conjugations acting via ordinary complex conjugation; and as infinite-dimensional numbers, with charge conjugations acting by automorphisms.     

Supersymmetry algebras that include topological charges

We show that in supersymmetric theories with solitons, the usual supersymmetry algebra is not valid; the algebra is modified to include the topological quantum numbers as central charges. Using the corrected algebra, we are able to show that in certain four dimensional gauge theories, there are no quantum corrections to the classical mass spectrum. These are theories for which Bogomolny has derived a classical bound; the argument involves showing that Bogomolny's bound is valid quantum mechanically and that it is saturated.