Duality in Interacting Particle Systems and Boson Representation

In the context of Markov processes, we show a new scheme to derive dual processes and a duality function based on a boson representation. This scheme is applicable to a case in which a generator is expressed by boson creation and annihilation operators. For some stochastic processes, duality relations have been known, which connect continuous time Markov processes with discrete state space and those with continuous state space. We clarify that using a generating function approach and the Doi-Peliti method, a birth-death process (or discrete random walk model) is naturally connected to a differential equation with continuous variables, which would be interpreted as a dual Markov process. The key point in the derivation is to use bosonic coherent states as a bra state, instead of a conventional projection state. As examples, we apply the scheme to a simple birth-coagulation process and a Brownian momentum process. The generator of the Brownian momentum process is written by elements of the SU(1,1) algebra, and using a boson realization of SU(1,1) we show that the same scheme is available.     

Spectral properties in a class of operators and group representations in nested Hilbert spaces

We characterize the spectrum of the elements of some operator algebras acting in a nested Hilbert space. In one of those algebras we prove the spectral theorem for Hermitian operators and the SNAG theorem for unitary representations of locally compact Abelian groups. Then we extend the SNAG theorem to representations in three more general classes of operators.     

Nilpotent locally convex Lie algebras and Lie field structures

The purpose of this work is to join Lie field structures with certain infinite-dimensional Lie algebras with locally convex topology. These topological Lie algebras allow topological groups which are a generalization of the connected nilpotent Lie groups. We showed the existence of the continuous unitary representations of the gained groups and then we proved the analogue of Gårding theorem. Using this theorem we established the existence of representations of Lie field structures into Lie algebras of skew-symmetric operators on Hilbert spaces.Work supported by National Science Foundation.On leave of absence from the Institute Rudjer Bokovi, Zagreb.     

Orthogonal polynomials in the lie algebras

We present a method of constructing orthogonal polynomials generated by pairs of Hermitian operators in representations of Lie algebras. All known classical polynomials of both discrete and continuous argument are generated naturally by the simplest Lie algebras.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 60–66, May, 1986.     

Global Geometric Deformations of Current Algebras as Krichever-Novikov Type Algebras

We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finite-dimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically not rigid, despite its formal rigidity. This shows that in the infinite dimensional Lie algebra case the relations between geometric deformations, formal deformations and Lie algebra two-cohomology are not that close as in the finite-dimensional case. The constructed families are e.g. of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory. The algebras are explicitly given by generators and structure equations and yield new examples of infinite dimensional algebras of current and affine Lie algebra type.     

Superalgebras,symplectic bosons and the Sugawara construction

New representations of affine Lie algebras are constructed using symplectic bosons of the sort that occur naturally in the BRST treatment of fermionic string theories. These representations are shown to have analogous properties to the current algebra representations in terms of free fermion fields, though they do not act in a positive space. In particular, the condition for the Sugawara construction of the Virasoro algebra to equal the free one is the existence of a superalgebra with a quadratic Casimir operator, paralleling the symmetric space theorem for fermionic field constructions. Both results are seen to be particular cases of a more general super-symmetric space theorem, which arises from considering an affinisation of the superalgebras. These algebras are realised in terms of free fermions and symplectic bosons and lead to a super-Sugawara construction of the Virasoro algebra. The conditions for this to equal a Virasoro algebra obtained from the free fields are provided by the super-symmetric space theorem.     

Composition of Lie Group Elements from Basis Lie Algebra Elements

It is shown explicitly how one can obtain elements of Lie groups as compositions of products of other elements based on the commutator properties of associated Lie algebras. Problems of this kind can arise naturally in control theory. Suppose an apparatus has mechanisms for moving in a limited number of ways with other movements generated by compositions of allowed motions. Two concrete examples are: (1) the restricted parallel parking problem where the commutator of translations in y and rotations in the xy-plane yields translations in x. Here the control problem involves a vehicle that can only perform a series of translations in y and rotations with the aim of efficiently obtaining a pure translation in x; (2) involves an apparatus that can only perform rotations about two axes with the aim of performing rotations about a third axis. Both examples involve three-dimensional Lie algebras. In particular, the composition problem is solved for the nine three- and four-dimensional Lie algebras with non-trivial solutions. Three different solution methods are presented. Two of these methods depend on operator and matrix representations of a Lie algebra. The other method is a differential equation method that depends solely on the commutator properties of a Lie algebra. Remarkably, for these distinguished Lie algebras the solutions involve arbitrary functions and can be expressed in terms of elementary functions.     

Coherent States on Lie Algebras: A Constructive Approach

We generalize the notion of coherent states toarbitrary Lie algebras by making an analogy with the GNSconstruction in C*-algebras. The method is illustratedwith examples of semisimple and nonsemisimple finite-dimensional Lie algebras as well as loopand Kac–Moody algebras. A deformed addition on theparameter space is also introduced simplifying someexpressions and some applications to conformal field theory are pointed out, e.g., differentialoperator and free field realizations found.     

Some compatible Poisson structures and integrable bi-Hamiltonian systems on four dimensional and nilpotent six dimensional symplectic real Lie groups

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi’s theorem we obtain new bi-Hamiltonian systems with four dimensional and nilpotent six dimensional symplectic real Lie groups as phase spaces.     

Discrete Approximations of Determinantal Point Processes on Continuous Spaces: Tree Representations and Tail Triviality

We prove tail triviality of determinantal point processes \( \mu \) on continuous spaces. Tail triviality has been proved for such processes only on discrete spaces, and hence we have generalized the result to continuous spaces. To do this, we construct tree representations, that is, discrete approximations of determinantal point processes enjoying a determinantal structure. There are many interesting examples of determinantal point processes on continuous spaces such as zero points of the hyperbolic Gaussian analytic function with Bergman kernel, and the thermodynamic limit of eigenvalues of Gaussian random matrices for \(\hbox {Sine}_2 , \hbox {Airy}_2 , \hbox {Bessel}_2 \), and Ginibre point processes. Our main theorem proves all these point processes are tail trivial.     

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     

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     

Unitary representations of some infinite-dimensional Lie algebras motivated by string theory on AdS3

We consider some unitary representations of infinite-dimensional Lie algebras motivated by string theory on AdS₃. These include examples of two kinds: the A,D,E type affine Lie algebras and the N=4 superconformal algebra. The first presents a new construction for free field representations of affine Lie algebras. The second is of a particular physical interest because it provides some hints that a hybrid of the NSR and GS formulations for string theory on AdS₃ exists.     

On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations

In the paper,we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly.By the approach the various loop algebras of the Lie algebra A_1are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained,respectively.A reduction of the later hierarchy is just right the famous Ablowitz-Ladik hierarchy.Finally,via two different enlarging Lie algebras of the Lie algebra A_1,we derive two resulting differential-difference integrable couplings of the Toda hierarchy,of course,they are all various discrete expanding integrable models of the Toda hierarchy.When the introduced spectral matrices are higher degrees,the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple.     

Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

We present an axiomatic formulation of a new class of infinitedimensional Lie algebras-the generalizations ofZ-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras continuum Lie algebras. The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered.     

Orthogonal Measures on State Spaces and Context Structure of Quantum Theory

An interplay between recent topos theoretic approach and standard convex theoretic approach to quantum theory is discovered. Combining new results on isomorphisms of posets of all abelian subalgebras of von Neumann algebras with classical Tomita’s theorem from state space Choquet theory, we show that order isomorphisms between the sets of orthogonal measures (resp. finitely supported orthogonal measures) on state spaces endowed with the Choquet order are given by Jordan ?-isomorphims between corresponding operator algebras. It provides new complete Jordan invariants for σ-finite von Neumann algebras in terms of decompositions of states and shows that one can recover physical system from associated structure of convex decompositions (discrete or continuous) of a fixed state.     

Fock Representations of Lie Algebras,Loop Algebras and Affine Lie Algebras

Explicit Fock representations of the classical Lie algebras in terms of boson creation and annihilation operators with an arbitrary highest weight are derived, and a general rule to construct Fock represen tations of a loop algebra from a boson realization ofits corresponding Lie algebra is establislted. A new kind of lowest weight represen tations of the affine Lie algebras attached to the classical Lie algebras, which require a zero center, is also presented. Based on these, a simple affinization procedure is given to obtain the Fock representations of level 1 of these affine Lie algebras.     

李超代数综述

本文将李超代数介绍给物理学工作者。讨论了阶化向量空间和李超代数的基本性质。给出经典李超代数的分类及基本经典李超代数的Kac-Dynkin图。特别是介绍了如何运用广义的不可约张量算符及Wigner-Eckart定理,求经典李超代数有限维不约可表示的方法。     

Iwasawa nilpotency degree of non compact symmetric cosets in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathcal{N} $\end{document}‐extended supergravity

We analyze the polynomial part of the Iwasawa realization of the coset representative of non compact symmetric Riemannian spaces. We start by studying the role of Kostant's principal SU(2)_P subalgebra of simple Lie algebras, and how it determines the structure of the nilpotent subalgebras. This allows us to compute the maximal degree of the polynomials for all faithful representations of Lie algebras. In particular the metric coefficients are related to the scalar kinetic terms while the representation of electric and magnetic charges is related to the coupling of scalars to vector field strengths as they appear in the Lagrangian. We consider symmetric scalar manifolds in ��‐extended supergravity in various space‐time dimensions, elucidating various relations with the underlying Jordan algebras and normed Hurwitz algebras. For magic supergravity theories, our results are consistent with the Tits‐Satake projection of symmetric spaces and the nilpotency degree turns out to depend only on the space‐time dimension of the theory. These results should be helpful within a deeper investigation of the corresponding supergravity theory, e.g. in studying ultraviolet properties of maximal supergravity in various dimensions.     

Symmetric Lie algebras of non-linear transformations of conformal type in quantum mechanics

The Koecher construction of simple symmetric Lie algebras is used to realize colineation and conformai Lie algebras of non-linear transformations of a pseudo-orthogonal vector space in the canonical Weyl algebras, which are used in the Schrödinger representation. The realization maps the linear sub-algebras onto symmetrized polynomials of second degree, whereas the non-linear parts are mapped onto polynomials of first and third degree. For the two examples the Meyberg Jordan algebras are explicitly given.