Wen-Li Yang, Yao-Zhong Zhang
We construct the explicit free field representations of the current algebras $so(2n)_k$, $so(2n+1)_k$ and $sp(2n)_k$ for a generic positive integer $n$ and an arbitrary level $k$. The corresponding energy-momentum tensors and screening currents of the first kind are also given in terms of free fields.
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The q-deformed super Virasoro algebra proposed by Chaichian and Presnajder is examined. Presented is the realizations by the FFZ algebra (the magnetic translation algebra) defined on a two-dimensional lattice with a supersymmetric Hamiltonian.
The fact that certain non-linear $W_{2,s}$ algebras can be linearized by the inclusion of a spin-1 current can provide a simple way to realize $W_{2,s}$ algebras from linear $W_{1,2,s}$ algebras. In this paper, we first construct the explicit field realizations of linear $W_{1,2,s}$ algebras with double-scalar and double-spinor, respectively. Then, after a change of basis, the realizations of $W_{2,s}$ algebras are presented. The results show that all these realizations are Romans-type realizations.
It has been shown that certain W algebras can be linearized by the inclusion of a spin-1 current. This Provides a way of obtaining new realizations of the W algebras. In this paper, we investigate the new ghost field realizations of the W(2,s)(s=3,4) algebras, making use of the fact that these two algebras can be linearized. We then construct the nilpotent BRST charges of the spinor non-critical W(2,s) strings with these new realizations.
Free Field Realization of $WBC_{n}$ and $WG_{2}$ algebras
研究了非单laced型仿射李代数量子Hamilton约化中的BRST-上同调。我们得到了$\bg=B_{2}$，$B_{3}$，$C_{3}$和$G_{2}$的$W{\bf g}$ -代数的自由场实现.通过对偶变换证明了$WC_{3}$代数与$WB_{3}$代数在量子水平上是等价的.
The V-algebras are the non-local matrix generalization of the well-known W-algebras. Their classical realizations are given by the second Poisson brackets associated with the matrix pseudodifferential operators. In this paper, by using the general Miura transformation, we give the decomposition theorems for the second Poisson brackets, from which we are able to construct the free field realizations for a class of V-algebras including V_{(2k,2)}-algebras that corresponds to the Lie algebra of C_k-type as the particular examples. The reduction of our discussion to the scalar case provides the si…
The property of some finite W-algebras to appear as the commutant of a particular subalgebra in a simple Lie algebra G is exploited for the obtention of new G-realizations from a "canonical" differential one. The method is applied to the conformal algebra so(4,2) and therefore yields also results for its Poincare subalgebra. Unitary irreducible representations of these algebras are recognized in this approach, which is naturally compared -or associated- to the induced representation technic.
Representations and Q-Boson Realization of the Algebra of Functions on the Quantum Group GLq(N)
本文详细研究了函数代数在量子群$ GL_q(n) $上的表示。一个q-analouge的根系构造这个代数，然后用于确定显式矩阵表示的生成元的这个代数。最后给出了$ GL_q(n) $生成元的q玻色子实现。
We give the realizations of the extended Weyl-Heisenberg (WH) algebra and the Rubakov-Spiridonov (RS) superalgebra in terms of anyons, characterized by the statistical parameter $ν\in[0,1]$, on two-dimensional lattice. The construction uses anyons defined from usual fermionic oscillators (Lerda-Sciuto construction). The anyonic realization of the superalgebra $sl(1/1)$ is also presented.
N = 2 Super $W_{\infty}$ Algebra and its Nonlinear Realization Through Super KP Formulation
通过对超KP族的一致的superLax表述，证明了超$W_{\infty}$代数的非线性实现的存在性. SuperLax算子的约化产生了由Inami和Kanno提出的$N=2$广义超KdV族的Lax算子。证明了Lax方程是Hamilton的，并给出了超场之间的Poisson括号代数，从而得到了非线性超$W_{\infty}$代数的实现.
The method of construction of Fock space realizations of Lie algebras is generalized for nonlinear algebras. We consider as an example the nonlinear algebra of constraints which describe the totally symmetric fields with higher spins in the AdS space-time.
Free Field Realization of $N=2$ Super $W_{3}$ Algebra
利用李超代数$A(2|1)$的超对称Miura变换得到的自由场实现，研究了量子$N=2$超- $W_{3}$代数。我们显式地计算了该代数的全算子乘积展开。计算结果与OPE方法所得结果一致。
Differential realizations in coordinate space for deformed Lie algebras with three generators are obtained using bosonic creation and annihilation operators satisfying Heisenberg commutation relations. The unified treatment presented here contains as special cases all previously given coordinate realizations of $so(2,1),so(3)$ and their deformations. Applications to physical problems involving eigenvalue determination in nonrelativistic quantum mechanics are discussed.
Free Superfield Realization of N=2 Quantum Super W_{3} Algebra
我们重写了一个非线性扩展的$N=2$超Virasoro代数--$ N=2$量子超$W_{3}$代数，它在$N=2$超空间中除了$1$维的超应力能量张量外，还含有一个额外的不带$U(1)$电荷的$2$维的主超场.该代数的自由超场实现是由两个$N=2$手征费米子超场得到的，其维数为$1/2$，满足$N=2$复$ U(1)$卡茨-穆迪代数.
Null Fields Realizations of $W_3$ from $W(sl(4),sl(3))$ and $W(sl(3|1),sl(3))$ Algebras
我们考虑了非线性代数$W(sl(4),sl(3))$和$W(sl(3|1),sl(3))$，并找到了它们的实现的电流跨越共形线性化代数。这些代数的特定结构，使我们能够构建实现模零域的$W_3$代数，位于陪集$W(sl(4),sl(3))/u(1)$和$W(sl(3|1),sl(3))/u(1)$。对于$W_3$代数中心电荷的以下值存在这样的实现：$c_W=-30,-40/7,-98/5,-2$。前两个值是第一次列出的，而对于其余的值，我们得到了新的实现方面的任意应力张量和$u(1)\times sl(2)$仿射电流。
In this paper, we investigate the spinor field realizations of the $W_{2,4}$ algebra, making use of the fact that the $W_{2,4}$ algebra can be linearized through the addition of a spin-1 current. And then the nilpotent BRST charges of the spinor non-critical $W_{2,4}$ string were built with these realizations.
Working at the level of Poisson brackets, we describe the extension of the generalized Wakimoto realization of a simple Lie algebra valued current, J, to a corresponding realization of a group valued chiral primary field, b, that has diagonal monodromy and satisfies $Kb'=Jb$. The chiral WZNW field b is subject to a monodromy dependent exchange algebra, whose derivation is reviewed, too.
Realizations of $κ$-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of $\mathfrak{gl}(n)$ generators. There are three one-parameter families of linear realizations for time-like and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between deformed Heisenberg algebra, star product, coproduct of momenta and twist operator is presented. It is proved that for each linear realization there exists Drinfeld twist…
We study Lie algebra $κ$-deformed Euclidean space with undeformed rotation algebra $SO_a(n)$ and commuting vectorlike derivatives. Infinitely many realizations in terms of commuting coordinates are constructed and a corresponding star product is found for each of them. The $κ$-deformed noncommutative space of the Lie algebra type with undeformed Poincar{é} algebra and with the corresponding deformed coalgebra is constructed in a unified way.
We present the explicit global realization of the isometries of anti-de Sitter like spaces of signature $(d_-,d_+)$, and their algebras in the space of functions on the pseudo-Riemannian symmetric space $SO(d_- +1,d_+) / SO(d_-,d_+)$. The process of going to the invariant boundaries leads to the realization of the corresponding conformal groups and algebras. We compute systematically the geodesics in these spaces by considering the coset representation of them.
Algebraic realizations of supersymmetry through SU(m,n) type superalgebras are developed. We show their applications to a bilocal quark-antiquark or a quark-diquark systems. A new scheme based on SU(6/1) is fully exploited and the bilocal approximation is shown to get carried unchanged into it. Color algebra based on octonions allows the introduction of a new larger algebra that puts quarks, diquarks and exotics in the same supermultiplet as hadrons and naturally suppresses quark configurations that are symmetrical in color space and antisymmetrical in remaining flavor, spin and position varia…
Reaction-Diffusion Processes as Physical Realizations of Hecke Algebras
我们表明，在一维的简单扩散和某些化学反应过程的动力学的主方程给出时间演化算子（哈密顿算子），这是Hecke代数的实现。在简单扩散的情况下，得到的相似变换后，可约厄米特表示，而在其他情况下，他们是非厄米特，并对应于超对称的Hecke代数。
Many Boson Realizations of Universal Nonlinear $W_{\infty}$-Algebras
利用具有$p-q=N$的不同签名的$p$加$q$标量，得到了与KP Hamilton结构相同的泛非线性$\hat{W}_{\infty}^{(N)}$（$\hat{W}_{1+\infty}^{(N)}$）代数的无限多个自由场实现.它们是三浦变换的推广，并通过后者的相应实现自然地产生修改的KP层次。它们的特征李代数起源被证明是分次的$SL(p,q)$。
Extended Superconformal Algebras and Free Field Realizations from Hamiltonian Reduction
我们发展了仿射李超代数的哈密尔顿约化方法，得到了经典和量子扩张超共形代数的显式和一般表达式。通过对对角规范和Drinfeld-Sokolov规范进行规范变换，并考虑量子修正，得到了该代数的经典和量子自由场实现的一般表达式.
We present a closed formula for a family of star-products by replacing the partial derivatives in the Moyal-Weyl formula with commuting vector fields. We show how to reproduce algebra relations on commutative spaces with these star-products and give some physically interesting examples of that procedure.
Nonlinear Realizations of the $W_3^{(2)}$ Algebra
本文考虑经典Polyakov代数$W_3^{(2)}$的非线性实现。利用陪集空间方法和协变约化过程，我们可以得到空间坐标和协变坐标互换的Boussinesq方程。通过增加一个空间坐标并引入$W_3^{(2)}$代数的两个副本，相同的方法产生$sl(3,R)$ Toda晶格方程。
Defining the $WBC_n$ algebras as the commutant of certain screening charges a special form for the classical generators is obtained which does not change under quantisation. This enables us to give explicitly the first few generators in a compact form for arbitrary $WBC_n$ algebras.
We study realizations of polynomial deformations of the sl(2,R)- Lie algebra in terms of differential operators strongly related to bosonic operators. We also distinguish their finite- and infinite-dimensional representations. The linear, quadratic and cubic cases are explicitly visited but the method works for arbitrary degrees in the polynomial functions. Multi-boson Hamiltonians are studied in the context of these ``nonlinear'' Lie algebras and some examples dealing with quantum optics are pointed out.
Non-Polynomial Realizations of W-Algebras
放松第一类约束条件，在通常的Drinfeld-Sokolov哈密顿约化导致对称固定后，实现W代数表示的所有J-电流分量。给出了非例外单（有限仿射）代数G的一般结果.这样的计算直接提供了幂零子代数$G_-$在G包络代数（的闭包）中的交换子，其中下标指G中所选的阶数。在仿射的情况下，明确的表达式Virasoro，$W_3$，Bershadsky代数在量子水平。
The superalgebra $\hat{K}'(4)$ and the exceptional N = 6 superconformal algebra have ``small'' irreducible representations in the superspaces $V^μ = t^μ\C[t, t^{-1}]\otimesΛ(N)$, where N = 2 and 3, respectively. For ${μ\in \C\backslash \Z}$ they are associated to the embeddings of these superalgebras into the Lie superalgebras of pseudodifferential symbols on the supercircle S^{1|N}. In this work we describe $\hat{K}'(4)$ and the exceptional N = 6 superconformal algebra in terms of matrices over a Weyl algebra. Correspondingly, we obtain realizations of their representations in $V^μ$ for $μ= 0…
A comment on the magical realizations of W_3
在用自由玻色子研究W_3的经典实现的过程中，Romans揭示了它与具有三次范数的有限维Jordan代数的关系。这些代数已被分类，并由一个无穷级数（产生“一般”实现）和四个与实除代数相关的零星代数（产生“神奇”实现）组成。一般的实现是由罗马人展示给萨缪尔森的，萨缪尔森没有解决魔法实现的量化问题。在后来的工作中，穆罕默迪表明，前两个神奇的实现无法在量子化中幸存下来。在这篇笔记中，我们通过证明其他两种神奇的实现也不存在来结束这个问题。
We give an explicit quantum super field construction of the N=2 super Casimir WA(n)-algebras, which is obtained from supersymmetric Miura transformation for the Lie superalgebra A(n,n-1). And also we give an extension of this algebra including a super vertex operator which depends on simple root system of A(n,n-1).
Anyonic Realization of $SU_q(N)$ Quantum Algebra
通过考虑二维晶格上的一组$N$非离子振子（介于费米子和玻色子振子之间的非定域、本征二维物体），我们利用广义Schwinger构造实现了$SU_q(N)$量子代数.我们发现该代数的形变参数$q$与任意子统计参数$ν$之间有$q=exp({\rm i}πν)$的关系。
We use superalgebras to realize the 3-algebras used to construct N=6, 8 Chern-Simons-matter (CSM) theories. We demonstrate that the superalgebra realization of the 3-algebras provides a unified framework for classifying the gauge groups of the N \geq 5 theories based on 3-algebras. Using this realization, we rederive the ordinary Lie algebra construction of the general N=6 CSM theory from its 3-algebra counterpart, and reproduce all known examples as well. In particular, we explicitly construct the Nambu 3-bracket in terms of a double graded commutator of PSU(2|2). The N = 8 theory of Bagger, …
The free field representation of the osp(1|2) current algebra is analyzed. The four point conformal blocks of the theory are studied. The structure constants for the product of an arbitrary primary operator and a primary field that transforms according to the fundamental representation of osp(1|2) are explicitly calculated.
We derive properties of N-extended GR super Virasoro algebras. These include adding central extensions, identification of all primary fields and the action of the adjoint representation on its dual. The final result suggest identification with the spectrum of fields in supergravity theories and superstring/M-theory constructed from NSR N-extended supersymmetric ${\cal {GR}}$ Virasoro algebras.
We find out that some unitary minimal models of the N=1 ${\cal SW}(3/2,2)$ superconformal algebra can be realized as the level one coset models based on the Wolf spaces $SU(n)/(SU(n-2)\times SU(2))$. We obtain the expression of the fermionic current with the conformal weight 5/2 in the algebra. Then, these models are twisted to give the topological conformal field theories.
We present a nonlinear realization of the 5-graded Lie algebra associated to a Kantor triple system. Any simple Lie algebra can be realized in this way, starting from an arbitrary 5-grading. In particular, we get a unified realization of the exceptional Lie algebras f_4, e_6, e_7, e_8, in which they are respectively related to the division algebras R, C, H, O.
Realizations of the $q$-Heisenberg and $q$-Virasoro Algebras
给出了$q$ -Heisenberg代数和$q$ -维拉索罗代数的场论实现.特别地，我们得到的电流和能量动量张量之间的算子乘积展开使用Sugawara建设。
Complex analytic realizations for quantum algebras
介绍了一类变形代数的复解析实现的方法，该方法基于它们各自的变形映射和它们的普通相干态。这种实现的明确结果提供的情况下的$q$ -振荡器（$q$ -Weyl-海森堡代数）和$su_{q}(2)$和$su_{q}(1,1)$代数和他们的共同产品。他们给出了一系列的权力的普通衍生运营商的行为上的巴格曼-希尔伯特空间的功能赋予通常的整合措施。在$q\rightarrow 1$极限下，这些实现归结为三个代数的通常解析Bargmann实现。
Explicit boundary conditions are given at spatial infinity for four-dimensional supergravity, which provide a realization of the super-BMS algebra of Awada, Gibbons and Shaw. The results are then generalized to the $N$- extended super-BMS algebras.
A realization of the affine Lie algebra ${A^{(1)}_1}$ and the relevant $Z$-algebra at negative level $-k$ is given in terms of parafermions. This generalizes the recent work on realization of the affine Lie algebra at the critical level.
We show that the elementary modes of the planar harmonic oscillator can be quantised in the framework of quantum mechanics based on pseudo-hermitian hamiltonians. These quantised modes are demonstrated to act as dynamical structures behind a new Jordan - Schwinger realization of the SU(1,1) algebra. This analysis complements the conventional Jordan - Schwinger construction of the SU(2) algebra based on hermitian hamiltonians of a doublet of oscillators.
Twisting of quantum (super)algebras. Connection of Drinfeld's and Cartan-Weyl realizations for quantum affine algebras
我们表明，一些因素的通用R-矩阵产生家庭的扭曲的标准的Hopf结构的任何量化contragradient李（超）代数的有限增长。作为应用，我们证明了任意两个具有不同Cartan矩阵的同构超代数具有同构的q-变形（作为结合超代数），并且它们的标准余乘是通过这种扭曲连接的。我们还提出了量子化仿射非扭曲卡茨-穆迪代数的第二Drinfeld实现的生成元与Cartan-Weyl生成元之间的显式关系。讨论了与这种同构密切相关的量子Cartan-Weyl基理论的进一步发展。证明了第二种实现的一个余乘的Drinfeld公式是标准的余乘被泛R-矩阵的因子扭曲的结果。最后，讨论了Drinfeld余乘的性质.
We study an N-body Calogero model in the S_N-symmetric subspace of the positive definite Fock space. We construct a new algebra of S_N-symmetric operators represented on the symmetric Fock space, and find a natural orthogonal basis by mapping the algebra onto the Heisenberg algebra. Our main result is the bosonic realization of nonlinear symmetry algebra describing the structure of degenerate levels of Calogero model.
In a vertex algebraic framework, we present an explicit description of the twisted Wakimoto realizations of the affine Lie algebras in correspondence with an arbitrary finite order automorphism and a compatible integral gradation of a complex simple Lie algebra. This yields generalized free field realizations of the twisted and untwisted affine Lie algebras in any gradation. The free field form of the twisted Sugawara formula and examples are also exhibited.
We unify kappa-Minkowki spacetime and Lorentz algebra in unique Lie algebra. Introducing commutative momenta, a family of kappa-deformed Heisenberg algebras and kappa-deformed Poincare algebras are defined. They are specified by the matrix depending on momenta. We construct all such matrices. Realizations and star product are defined and analyzed in general and specially, their relation to coproduct of momenta is pointed out. Hopf algebra of the Poincare algebra, related to the covariant realization, is presented in unified covariant form. Left-right dual realizations and dual algebra are intr…
It is shown that the Gel'fand-Tsetlin realization of irreducible representations of the $A_n$ algebra is directly connected with a linear exactly integrable system in the n-dimensional space. General solution for this system is explicitly given.
The $gl(4|4)$ current algebra at general level $k$ is investigated. Its free field representation and corresponding energy-momentum tensor are constructed. Seven screening currents of the first kind are also presented.
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.
Anyonic Realization of the Quantum Affine Lie Algebra U_q(A_N)
给出了量子仿射李代数$U_q(\hat A_{N-1})$在二维格点上任意子的一种实现，形变参数$q$与任意子的统计参数$ν$通过$q = e^{iπν}$联系起来.在变形参数趋于1的极限下，我们恢复了未变形仿射李代数的Feingold-Frenkel费米子结构。
The method of nonlinear realizations is applied to the l-conformal Galilei algebra to construct a dynamical system without higher derivative terms in the equations of motion. A configuration space of the model involves coordinates, which parametrize particles in d spatial dimensions, and a conformal mode, which gives rise to an effective external field. It is shown that trajectories of the system can be mapped into those of a set of decoupled oscillators in d dimensions.
We construct an irreducible representation for the extended affine algebra of type $sl_2$ with coordinates in a quantum torus. We explicitly give formulas using vertex operators similar to those found in the theory of the infinite rank affine algebra $A_{\infty}$.
In the Dirac bracket approach to dynamical systems with second class constraints observables are represented by elements of a quotient Dirac bracket algebra. We describe families of new realizations of this algebra through quotients of the original Poisson algebra. Explicite expressions for generators and brackets of the algebras under consideration are found.
We give a realization of the quantum affine Lie algebras $\uqa$ and $\uqc$ in terms of anyons defined on a one-dimensional chain (or on a two-dimensional lattice), the deformation parameter $q$ being related to the statistical parameter $ν$ of the anyons by $q = e^{iπν}$. In the limit of the deformation parameter going to one we recover the Feingold-Frenkel fermionic construction of undeformed affine Lie algebras.
Starting from the well-known quantum Miura-like transformation for the non simply-laced Lie algebras B(3),we give an explicit construction of the Casimir WB(3) algebras.We reserve the notation WB(N) for the Casimir W algebras of type W(2,4,6,...,2N,N+1/2) which contains one fermionic field. It is seen that WB(3) algebra is closed an associative for all values of the central element c.
The $osp(2n|2n)$ current algebra for a {\it generic} positive integer $n$ at general level $k$ is investigated. Its free field representation and corresponding energy-momentum tensor are constructed. The associated screening currents of the first kind are also presented.
We present Lie-algebraic deformations of Minkowski space with undeformed Poincaré algebra. These deformations interpolate between Snyder and $κ$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. By introducing modules, it is shown that although deformed and undeformed structures are not isomorphic at the level of vector spaces, they are however isomorphic at the level of Hopf algebraic action on corresponding modules. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $κ$-deformed S…
In this paper we construct strong exceptional collections of vector bundles on smooth projective varieties that have a prescribed endomorphism algebra. We prove the construction problem always have a solution. We consider some applications to noncommutative projective planes and to the quiver connected with the 3-point Ising function.
It is well-known that the chiral WZNW Bloch waves satisfy a quadratic classical exchange algebra which implies the affine Kac-Moody algebra for the corresponding currents. We here obtain a direct derivation of the exchange algebra by inverting the symplectic form on the space of Bloch waves, and give a completely algorithmic construction of its generalized free field realizations that extend the classical Wakimoto realizations of the current algebra.
In a Hamiltonian system with first class constraints observables can be defined as elements of a quotient Poisson bracket algebra. In the gauge fixing method observables form a quotient Dirac bracket algebra. We show that these two algebras are isomorphic. A new realization of the observable algebras through the original Poisson bracket is found. Generators, brackets and pointwise products of the algebras under consideration are calculated.
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