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.     

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.     

On a construction of the maximal additive field of v. Neumann algebras in quantum field theory

The construction presented below shows how an algebra without the additive property (called shortly a non-additive algebra) can be reduced to an additive algebra. The algebra obtained this way is the largest additive algebra contained in the original non- additive algebra. Additivity is one of the features postulated for the observable algebra in Haag's field theory (algebras of this type, belonging to a special class, are called by Haag kinematical ones). From another way it is well known that there are many algebras without such property (e.g. the algebra of fermions).     

Proper Effect Algebras Admitting No States

We show that there is even a finite proper effect algebra admitting no states. Further, every lattice effect algebra with an ordering set of valuations is an MV effect algebra (consequently it can be organized into an MV algebra). An example of a regular effect algebra admitting no ordering set of states is given. We prove that an Archimedean atomic lattice effect algebra is an MV effect algebra iff it admits an ordering set of valuations. Finally we show that every nonmodular complete effect algebra with trivial center admits no order-continuous valuations.     

Irreducible representations of Virasoro-toroidal Lie algebras

Toroidal Lie algebras and their vertex operator representations were introduced in [MEY] and a class of indecomposable modules were investigated. In this work, we extend the toroidal algebra by the Virasoro algebra thus constructing a semi-direct product algebra containing the toroidal algebra as an ideal and the Virasoro algebra as a subalgebra. With the use of vertex operators and certain oscillator representations of the Virasoro algebra it is proved that the corresponding Fock space gives rise to a class of irreducible modules for the Virasoro-toroidal algebra.To A. John Coleman on the occasion of his 75th birthday     

Dirac matrices as elements of a superalgebraic matrix algebra

A Clifford extension of the Grassmann algebra is considered in which operators are built from products of Grassmann variables and derivatives with respect to them. It is shown that a subalgebra of operators, isomorphic to the usual matrix algebra, can be separated in this algebra, while the algebra itself is a generalization of the matrix algebra, contains superalgebraic operators expanding the matrix algebra, and produces transformations of supersymmetry.     

Lie algebra type noncommutative phase spaces are Hopf algebroids

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.     

Differential algebra of quantum fermions

Lie coalgebra equips an exterior algebra (algebra of fermions) with a structure of a differential algebra. In similar way we equip an algebra of quantum fermions (quantized exterior algebra) with a structure of a differential algebra. This leads to a notion of a variety of Lie coalgebras for a Hecke braid. This approach is different from that of Gurevich (1988 and 1993), Woronowicz (1989) and of Majid (1993).     

Construction of a left Hilbert algebra with respect to a Minkowski form

We construct a left Hilbert algebra with respect to a Minkowski form and generalize the theorem that every von Neumann algebra is isomorphic to the left von Neumann algebra of a left Hilbert algebra.     

q-deformed Lorentz-algebra in Minkowski phase space

In the present paper we show that the Lorentz algebra as defined in [5] is isomorphic to an algebra closely related to a q-deformed algebra. On this algebra we define a Hopf algebra structure and show its action on q-spinor modules. This algebra is related to the q-deformed Minkowski space algebra by a non invertible factorisation. Received: 12 June 1998 / Published online: 5 October 1998     

Asymptotic symmetry of geometries with Schrödinger isometry

We show that the asymptotic symmetry algebra of geometries with Schrödinger isometry in any dimension is an infinite-dimensional algebra containing one copy of Virasoro algebra. It is compatible with the fact that the corresponding geometries are dual to non-relativistic CFTs whose symmetry algebra is the Schrödinger algebra which admits an extension to an infinite-dimensional symmetry algebra containing a Virasoro subalgebra.     

Applications of Polynomial Algebras to 2-Dimensional Deformed Oscillators

The polynomial algebra is a deformed su(2) algebra. Here, we use polynomial algebra as a method to solve a series of deformed oscillators. Thus, we find a series of physics systems corresponding with polynomial algebra with different highest orders.     

Noncommutative differential geometry, and the matrix representations of generalised algebras

The underly ing algebra I or a noncummutative geometry is taken to be a matrix algebra, and the set of derivatives the ad joint of a subset of traceless matrices. This is sufficient to calculate the dual 1-forms, and show that the space of 1-firms is at free module over the algebra of matrices. The concept of a generalised algebra is delined and it is shown that this is required in order for the space of 2-forms to exist, The exterior derivative is generalised for higher-order forms and these are also shown to he free modules over the matrix algebra. Examples of mappings that preserve the differential Structure are peen, Also giken are four examples of matrix generalised algebras, and the corresponding noncommutntive geometries, including the cases where the generalised algebra corresponds to a representation of a Lie algebra or a q-deformed algebra.     

Algebraical comparison of classical and quantum polynomial observables

The symplectic vector spaceE of theq andp's of classical mechanics allows a basis free definition of the Poisson bracket in the symmetric algebra overE. Thus the symmetric algebra overE becomes a Lie algebra, which can be compared with the quantum mechanical Weyl algebra with its commutator Lie structure. The universality of the Weyl algebra is used to study the well-known ‘classical’ Moyal realisation of the Weyl algebra in the symmetric algebra. Quantisations are defined as linear mappings of the underlying vector spaces of the two algebras. It is shown that the classical Lie algebra is −2 graded, whereas the quantum Lie algebra is not. This proves that they are not isomorphic, and hence there is no Dirac quantisation.     

Boson realizations of quantum algebras and some physical spaces with quantum algebra symmetry

The generators ofq-boson algebra are expressed in terms of those of boson algebra, and the relations among the representations of a quantum algebra onq-Fock space, on Fock space, and on coherent state space are discussed in a general way. Two examples are also given to present concrete physical spaces with quantum algebra symmetry. Finally, a new homomorphic mapping from a Lie algebra to boson algebra is presented.This work is supported by the National Foundation of Natural Science of China.     

Lax表示的Kac—Moody代数结构

本文指出,非线性(演化)系统的广义Lax表示所取值的代数,即延拓代数y×D(λ),实质上是Kac-Moody代数。这里,y是一有限维李代数,D(λ)是谱参数λ的值域。本文并利用Dolan关于主手征模型Kac-Moody代数的实现,给出了一类1+1维非线性(演化)系统的Kac-Moody代数的实现。 关键词：     

A van Est Isomorphism for Bicrossed Product Hopf Algebras

To any locally finite representation of a given double crossed sum (product) Lie algebra (group), we associate a stable anti Yetter-Drinfeld (SAYD) module over the bicrossed product Hopf algebra which arises from the semidualization procedure. We prove a van Est isomorphism between the relative Lie algebra cohomology of the total Lie algebra and the Hopf cyclic cohomology of the corresponding Hopf algebra with coefficients in the associated SAYD module.     

Realization of Dirac Bracket Algebras Through First-Class Functions and Quantization of Constrained Systems

It is shown that a Dirac bracket algebra is isomorphic to the original Poisson bracket algebra of first-class functions subject to first-class constraints. The isomorphic image of the Dirac bracket algebra in the star-product commutator algebra is found.     

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.     

Leibniz and Lie Algebra Structures for Nambu Algebra

We canonically associate a Leibniz algebra with every Nambu algebra. We show how various homological and cohomological complexes for a Nambu algebra can be naturally obtained from its structure as a module over the Leibniz algebra. We also present a generalization of a classical Lie--Berezin construction for Nambu algebras and extend these results for Nambu superalgebras.