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).     

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.     

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.     

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).     

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.     

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     

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.     

Embedding Q-deformed Heisenberg algebras into undeformed ones

Any deformation of a Weyl or Clifford algebra can be realized through a change of generators in the undeformed algebra. Here we briefly describe and motivate our systematic procedure for constructing all such changes of generators for those particular deformations where the original algebra is covariant under some Lie group and the deformed algebra is covariant under the corresponding quantum group.     

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.     

Yang–Mills Algebra

Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang–Mills connection over the (s+1)-dimensional pseudo Euclidean space are pointed out. This algebra is Koszul of global dimension 3 and Gorenstein but except for s=1 (i.e. in the two-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin–Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincaré series of this algebra and for the dimension in degree n of the graded Lie algebra of which is the universal enveloping algebra. In the four-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one.     

The blob algebra and the periodic Temperley-Lieb algebra

We determine the structure of two variations on the Temperley-Lieb algebra, both used for dealing with special kinds of boundary conditions in statistical mechanics models. The first is a new algebra, the blob algebra. We determine both the generic and all the exceptional structures for this two parameter algebra. The second is the periodic Temperley-Lieb algebra. The generic structure and part of the exceptional structure of this algebra have already been studied. We complete the analysis using results from the study of the blob algebra.     

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.     

Local symmetries of massless wave equations

The structure of a wave-equation symmetry algebra for massless particles is studied. It is shown that the local-symmetry algebra of massless wave equations belongs to the enveloping algebra of a conformal-group Lie algebra. It is also proven that the space of nontrivial symmetries of fixed order is irreducible under adjoint representation of the conformal algebra, and its dimension is indicated. Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 19–26, May, 1998.     

第一类P?schl-Teller势的非线性代数结构

在形变李代数理论的基础上,利用哈密顿算符和自然算符,构造出第一类P?schl-Teller势的非线性谱生成代数.该非线性代数能够完全确定势场的能量本征态集合和本征值谱,在适当的非线性算符变换下可以化为谐振子代数,显示了该系统具有新的对称性 关键词： P?schl-Teller势 自然算符 非线性谱生成代数     

A three-parameter Hopf deformation of the algebra of Feynman-like diagrams*

We construct a three-parameter deformation of the Hopf algebra LDIAG. This is the algebra that appears in an expansion in terms of Feynman-like diagrams of the product formula in a simplified version of quantum field theory. This new algebra is a true Hopf deformation which reduces to LDIAG for some parameter values and to the algebra of matrix quasi-symmetric functions (MQSym) for others, and thus relates LDIAG to other Hopf algebras of contemporary physics. Moreover, there is an onto linear mapping preserving products from our algebra to the algebra of Euler–Zagier sums.     

Twisted Quantum Toroidal Algebras {T_q^-(\mathfrak g)}

We construct a principally graded quantum loop algebra for the Kac–Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.     

第一类Poschl-Teller势的非线性代数结构

在形变李代数理论的基础上,利用哈密顿算符和自然算符,构造出第一类Poschl-Teller势的非线性谱生成代数。该非线性代数能够完全确定势场的能量本征态集合和本征值谱,在适当的非线性算符变换下可以化为谐振子代数,显示了该系统具有新的对称性。     

Non-linear realization of the virasoro-Kac-Moody algebra and the anomalies

The non-linear realization of the Virasoro algebra ⊛ Kac-Moody algebra will be studied. We will calculate the Ricci tensor of the relevant Kähler manifold to show a new vacuum structure for this coupled algebra.     

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.     

A calculus based on a q-deformed Heisenberg algebra

We show how one can construct a differential calculus over an algebra where position variables x and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra has a subalgebra generated by x and its inverse which we call the coordinate algebra. A physical field is considered to be an element of the completion of this algebra. We can construct a derivative which leaves invariant the coordinate algebra and so takes physical fields into physical fields. A generalized Leibniz rule for this algebra can be found. Based on this derivative differential forms and an exterior differential calculus can be constructed. Received: 26 November 1998 / Published online: 27 April 1999