The number of independent traces and supertraces on symplectic reflection algebras

It is shown that A:= H₁, _η (G), the sympectic reflection algebra over ?, has T_G independent traces, where T_G is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group G ? Sp(2N) ? End(?^2N) generated by the system of symplectic reflections.

Simultaneously, we show that the algebra A, considered as a superalgebra with a natural parity, has S_G independent supertraces, where S_G is the number of conjugacy classes of elements without eigenvalue -1 belonging to G.

We consider also A as a Lie algebra A^L and as a Lie superalgebra A^S.

It is shown that if A is a simple associative algebra, then the supercommutant [A^S, A^S] is a simple Lie superalgebra having at least S_G independent supersymmetric invariant non-degenerate bilinear forms, and the quotient [A^L, A^L]/([A^L, A^L] ∩ ?) is a simple Lie algebra having at least T_G independent symmetric invariant non-degenerate bilinear forms.     

Supertrace and Superquadratic Lie Structure on the Weyl Algebra,and Applications to Formal Inverse Weyl Transform

Using the Moyal *-product and orthosymplectic supersymmetry, we construct a natural nontrivial supertrace and an associated nondegenerate invariant supersymmetric bilinear form for the Lie superalgebra structure of the Weyl algebra W. We decompose adjoint and twisted adjoint actions. We define a renormalized supertrace and a formal inverse Weyl transform in a deformation quantization framework and develop some examples Mathematics Subject Classification: 53D55, 17B05, 17B10, 17B20, 17B60, 17B65     

Nonlinear Deformed Algebra Realized in a Physical System

We show that nonlinear deformed algebra can exist in a physical system with Poschl-Teller potential. Due to this algebra, the eigenvalue problem of the system can be exactly solved by operator method. The raising and lowering operators satisfying this algebra are constructed. And the physical meaning of two deforming functions involving in this algebra is given. In addition, the SU(1,1) symmetry is exhibited in such a system by the operator method.     

On the q-Deformation of Heisenberg Algebra

This paper deals with a class of q-deformations of Heisenberg algebra which contains the q-Heisenberg algebra, the q-oscillator algebra and others. Their representation theory is considered for q being generic or a root of 1. Finally, the structure of Hopf algebra in a quotient algebra is also discussed.     

Algebra of Observables and Charge Superselection Sectors for QED on the Lattice

Quantum Electrodynamics on a finite lattice is investigated within the hamiltonian approach. First, the structure of the algebra of lattice observables is analyzed and it is shown that the charge superselection rule holds. Next, for every eigenvalue of the total charge operator a canonical irreducible representation is constructed and it is proved that all irreducible representations corresponding to a fixed value of the total charge are unique up to unitary equivalence. The physical Hilbert space is by definition the direct sum of these superselection sectors. Finally, lattice quantum dynamics in the Heisenberg picture is formulated and the relation of our approach to gauge fixing procedures is discussed. Received: 21 October 1996 / Accepted: 10 February 1997     

Single Boson Realizations of the Higgs Algebra

We obtained for the Higgs algebra three kinds of single boson realizations such as the unitary Holstein-Primakoff-like realization, the non-unitary Dyson-like realization, and the unitary Villain-like realization. The corre-sponding similarity transformations between the Holstein-Primakoff-like realizations and the Dyson-like realizations are given.     

q-Deformation of the Krichever–Novikov Algebra

Using q-operator product expansions between U(1) current fields and the corresponding energy-momentum tensors, we furnish the q-analogues of the generalized Heisenberg algebra and the Krichever–Novikov algebra.     

基于几何代数的彩色光流场计算

提出了一种新的基于几何代数的彩色光流场计算方法.从新的角度出发讨论了运用几何代数的概念来解决彩色图像序列的可行性和简便性.在几何代数域内对彩色图像序列用多重矢量表示,证明了彩色光流场分析时进行约束条件时的测量问题,通过多重矢量而非多通道的图像处理方法来表示图像连续特性,从而扩大了测量范围;然后依据物体色彩在运动过程中保持不变的原理,在几何代数域内推导了彩色图像序列的光流约束方程及其解法.结果表明,该算法能够显著提高各种情况下的彩色光流场计算能力.     

Single Boson Realizations of the Higgs Algebra

We obtained for the Higgs algebra three kinds of single boson realizations such as the unitary Holstein-Primakoff-like realization, the non-unitary Dyson-like realization, and the unitary Villain-like realization. The corre-sponding similarity transformations between the Holstein-Primakoff-like realizations and the Dyson-like realizations are given.     

Quantum Group Structure of the q-Deformed Virasoro Algebra

In this Letter, the quantum group structure of the q-deformed Virasoro algebra Vir_q will be given.     

Aspects of Generalized Calogero Model

A multispecies model of Calogero type in D 1 dimensions is constructed. The model includes harmonic, two-body and three-body interactions. Using the underlying conformal SU(1,1) algebra, we find the exact eigenenergies corresponding to a class of the exact global collective states. Analyzing corresponding Fock space, we detect the universal critical point at which the model exhibits singular behavior.     

Renormalized Traces and Cocycles on the Algebra of S 1-Pseudo-differential Operators

Using renormalized (or weighted) traces of classical pseudo-differential operators and calculus on formal symbols. We exhibit three cocycles on the Lie algebra of classical pseudo-differential operators $Cl(S^1,\mathbb{C}^n)Using renormalized (or weighted) traces of classical pseudo-differential operators and calculus on formal symbols. We exhibit three cocycles on the Lie algebra of classical pseudo-differential operators acting on . We first show that the Schwinger functional associated to the Dirac operator is a cocycle on , and not only on a restricted algebra Then, we investigate two bilinear functionals and , which satisfies We show that and are two cocycles in , and and have the same nonvanishing cohomology class. We finaly calculate on classical pseudo-differential operators of order 1 and on differential operators of order 1, in terms of partial symbols. By this last computation, we recover the Virasoro cocyle and the K?hler form of the loop group. Mathematics Subject Classification (1991). 47G30, 47N50     

Automorphisms of the Weyl Algebra

We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. Several arguments in favor of this conjecture are presented, all based on the consideration of the reduction of the Weyl algebra to positive characteristic Mathematics Subject Classification (2000) 13N10, 16S32, 16H05     

Kontsevich Star Product on the Dual of a Lie Algebra

We show that on the dual of a Lie algebra g of dimension d, the star product recently introduced by M. Kontsevich is equivalent to the Gutt star product on g*. We give an explicit expression for the operator realizing the equivalence between these star products.     

Expansion of Lie Algebra and Its Application

Firstly we expand a finite-dimensional Lie algebra into a higher-dimenslonal one. By making use of the later and its corresponding loop algebra, the expanding integrable model of the multi-component NLS-mKdV hierarchy is worked out.     

A New Quantum Analog of the Brauer Algebra

We introduce a new algebra depending on two nonzero complex parameters z and q such that its specialization at z=q ⁿ and q=1 coincides the Brauer algebra. We show that the action of the new algebra commutes with the representation of the twisted deformation of the enveloping algebra U(o _n) in the tensor power of the vector representation.     

On the Structure of the Observable Algebra for QED on the Lattice

We prove that the matter field subalgebra of the observable algebra for QED on a finite lattice is isomorphic to the enveloping algebra of the Lie algebra sl(2N, C), factorized by a certain ideal. Using this result, we give a new proof of the decomposition of the physical Hilbert space into charge superselection sectors.     

Deformation Quantizations of the Poisson Algebra of Laurent Polynomials

It is well known that the Moyal bracket gives a unique deformation quantization of the canonical phase space R2n up to equivalence. In his presentation of an interesting deformation quantization of the Poisson algebra of Laurent polynomials, Ovsienko discusses the equivalences of deformation quantizations of these algebras. We show that under suitable conditions, deformation quantizations of this algebra are equivalent. Though Ovsienko showed that there exists a deformation quantization of the Poisson algebra of Laurent polynomials which is not equivalent to the Moyal product, this is not correct. We show this equivalence by two methods: a direct construction of the intertwiner via the star exponential and a more standard approach using Hochschild 2-cocycles.