交换超算符方法的李代数研究

本文讨论了交换超算符方法的理论基础,结果表明由交换超算符所定义的算符集合g是一个李代数,交换超算符的定义就是李代数中内导子的定义,由此得出一些交换超算符间的代数关系。证明了g中所有算符诱导的超算符集合也是一个李代数,指出了与g对应的是由复盖群派生的,有内积定义的李群,而角动量超算符是由矢量场的内禀角动量和单位算符的直积所生成。结论是交换超算符方法的理论基础是李代数。     

Kostka matrices at the level of bases and the complete set of commuting Jucys-Murphy operators

A method for evaluation of Kostka matrices at the level of bases, and determination of related irreducible basis of the Weyl duality is proposed. The method bases on Jucys-Murphy operators which constitute a complete set of commuting Hermitian operators along the general Dirac formalism of quantum mechanics, applied to the algebra of a symmetric group. The way of construction of appropriate projection operators is pointed out, and the combinatorial meaning of the path on the Young graph, corresponding to a standard Young tableau, is made transparent.     

FRT presentation of classical Askey–Wilson algebras

Automorphisms of the infinite-dimensional Onsager algebra are introduced. Certain quotients of the Onsager algebra are formulated using a polynomial in these automorphisms. In the simplest case, the quotient coincides with the classical analog of the Askey–Wilson algebra. In the general case, generalizations of the classical Askey–Wilson algebra are obtained. The corresponding class of solutions of the non-standard classical Yang–Baxter algebra is constructed, from which a generating function of elements in the commutative subalgebra is derived. We provide also another presentation of the Onsager algebra and of the classical Askey–Wilson algebras.

     

Algebra of differential operators associated with Young diagrams

We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues are expressed through the characters of symmetric groups. The structure constants of the algebra are expressed through the Hurwitz numbers.     

On Ternary Quotients of Cubic Hecke Algebras

We prove that the quotient of the group algebra of the braid group introduced by Funar (Commun Math Phys 173:513–558, 1995) collapses in characteristic distinct from 2. In characteristic 2 we define several quotients of it, which are connected to the classical Hecke and Birman-Wenzl-Murakami quotients, but which admit in addition a symmetry of order 3. We also establish conditions on the possible Markov traces factorizing through it.     

Distances in finite spaces from noncommutative geometry

Following the general principles of noncommutative geometry, it is possible to define a metric on the space of pure states of the noncommutative algebra generated by the coordinates. This metric generalizes the usual Riemannian one. We investigate some general properties of this metric in finite commutative cases corresponding to a metric on a finite set, and also compute explicitly some distances associated to commutative or noncommutative algebras.     

Quantum mechanics on the noncommutative torus

We analyze the algebra of observables of a charged particle on a noncommutative torus in a constant magnetic field. We present a set of generators of this algebra which coincide with the generators for a commutative torus but at a different value of the magnetic field, and demonstrate the existence of a critical value of the magnetic field for which the algebra reduces. We then obtain the irreducible representations of the algebra and relate them to noncommutative bundles. Finally we comment on Landau levels, density of states and the critical case.     

A 9× 9 Matrix Representation of Birman-Wenzl-Murakami Algebra and Berry Phase in Yang-Baxter System

We present a 9×9 S-matrix and E-matrix. A representation of specialized Birman-Wenzl-Murakami algebra is obtained. Starting from the given braid group representation S-matrix, we obtain the trigonometric solution of Yang-Baxter equation. A unitary matrix \breve{R}(x,φ₁,φ₂) is generated via theYang-Baxterization approach. Then we construct a Yang-BaxterHamiltonian through the unitary matrix \breve{R}(x,φ₁,φ₂). Berry phase of this Yang-Baxter system is investigated in detail.     

Pre-Poisson Algebras

A definition of pre-Poisson algebras is proposed, combining structures of pre-Lie and zinbiel algebra on the same vector space. It is shown that a pre-Poisson algebra gives rise to a Poisson algebra by passing to the corresponding Lie and commutative products. Analogs of basic constructions of Poisson algebras (through deformations of commutative algebras, or from filtered algebras whose associated graded algebra is commutative) are shown to hold for pre-Poisson algebras. The Koszul dual of pre-Poisson algebras is described. It is explained how one may associate a pre-Poisson algebra to any Poison algebra equipped with a Baxter operator, and a dual pre-Poisson algebra to any Poisson algebra equipped with an averaging operator. Examples of this construction are given. It is shown that the free zinbiel algebra (the shuffle algebra) on a pre-Lie algebra is a pre-Poisson algebra. A connection between the graded version of this result and the classical Yang–Baxter equation is discussed.     

Framed Vertex Operator Algebras, Codes and the Moonshine Module

For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge ?, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge ? are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras. Received: 14 July 1997 / Accepted: 8 September 1997     

Generating function for extended Jacobi polynomials, noncommutative differential calculus and the relativistic energy and momentum operators

It is shown that the generating function for the matrix elements of irreps of Lorentz group is the common eigenfunction of the interior derivatives of the noncommutative differential calculus over the commutative algebra generated by the coordinate functions. These derivatives commute and can be interpreted as the quantum mechanical operators of the relativistic momentum corresponding to the half of the non-Euclidean distance from the origin in the Lobachevsky momentum space.     

Subalgebras of the algebra of a conformal group with nontrivial center

A similarity relationship is set up in the set of subalgebras with a nontrivial center of a conformal group algebra that separates it into classes of mutually similar subalgebras. In all 24 different classes of similar subalgebras are obtained and one representative is described in each class. All the nonequivalent two-dimensional commutative subalgebras of a conformal group algebra are obtained as an illustration of applying such a subalgebra system. The results of the research can be used to obtain the solutions of conformally invariant equations in the separated variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 45–48, May, 1990.     

Relativistic kinetic momentum operators, half-rapidities and noncommutative differential calculus

It is shown that the generating function for the matrix elements of irreps of Lorentz group is the common eigenfunction of the interior derivatives of the noncommutative differential calculus over the commutative algebra generated by the coordinate functions in the Relativistic Configuration Space (RCS). These derivatives commute and can be interpreted as the quantum mechanical operators of the relativistic momentum corresponding to the half of the non-Euclidean distance in the Lobachevsky momentum space (the mass shell).     

彩色图像处理的可交换 Clifford 代数方法

采用可交换Clifford代数对彩色图像建模,充分利用彩色图像作为一个整体所具有的潜在颜色信息,实现彩色图像各颜色分量的并行处理,可完成彩色图像的整体处理。本文分析了彩色图像的表示方法,系统研究了一类可交换Clifford代数-Clcom2,定义了Clcom2上元素的四则运算规则、单位元、逆元、共轭、范数等。给出了基于可交换Clifford代数的彩色图像表示方法,并介绍了一个Clcom2架构下的彩色图像处理实例：彩色图像边缘检测。与传统的四元数彩色图像表示方法相比,本文所提出的方法最大限度地去除了数据冗余,其算法复杂度也大大降低。结果显示,基于可交换Clifford代数的彩色图像表示方法可以应用到彩色图像处理中。     

Some Results on Novikov–Poisson Algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. A Novikov–Poisson algebra is a Novikov algebra with a compatible commutative associative algebraic structure, which was introduced to construct the tensor product of two Novikov algebras. In this paper, we commence a study of finite-dimensional Novikov–Poisson algebras. We show the commutative associative operation in a Novikov–Poisson algebra is a compatible global deformation of the associated Novikov algebra. We also discuss how to classify Novikov–Poisson algebras. And as an example, we give the classification of 2-dimensional Novikov–Poisson algebras.     

Atomic Sequential Effect Algebras

Various conditions ensuring that a sequential effect algebra or the set of sharp elements of a sequential effect algebra is a Boolean algebra are presented.     

On poisson structure and curvature

We consider a curved space-time whose algebra of functions is the commutative limit of a noncommutative algebra and which has therefore an induced Poisson structure. In a simple example we determine a relation between this structure and the Riemann tensor.     

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.     

Necessary and sufficient conditions for integral representations of Wightman functionals at Schwinger points

Given a set of Wightman functions one would like to associate to it a field on Euclidean space admitting a simultaneous diagonalization. We investigate when this can be done in such a way that the Schwinger functions are the expectation values of this commutative field with a bounded metric operator commuting with the field. This requires as a tool the characterization of those linear functionals on the symmetric tensor algebra over a space of test functions which can be represented by complex measures on the corresponding space of distributions.     

Duality on the quantum space(3) with two parameters

In this study, we introduce a dual Hopf algebra in the sense of Sudbery for the quantum space(3) whose coordinates satisfy the commutation relations with two parameters and we show that the dual algebra is isomorphic to the quantum Lie algebra corresponding to the Cartan-Maurer right invariant differential forms on the quantum space(3). We also observe that the quantum Lie algebra generators are commutative as those of the undeformed Lie algebra and the deformation becomes apparent when one studies the Leibniz rules for the generators.