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We find the free field construction of the basic W-current and screening currents for the deformed W-superalgebra W q , t A ( M , N ) associated with Lie superalgebra of type A(M, N). Using this free field construction, we introduce the higher W-currents and obtain a closed set of quadratic relations among them. These relations are independent of the choice of Dynkin diagrams for the Lie superalgebra A(M, N), though the screening currents are not. This allows us to define W q , t A ( M , N ) by generators and relations.

We revisit the construction of conformal field theories based on Takiff algebras and superalgebras that was introduced by Babichenko and Ridout. Takiff superalgebras can be thought of as truncated current superalgebras with Z-grading which arise from taking p copies of a Lie superalgebra g and placing them in the degrees s=0,...,p-1. Using suitably defined non-degenerate invariant forms we show that Takiff superalgebras give rise to families of conformal field theories with central charge c=p sdim(g). The resulting conformal field theories are defined in the standard way, i.e. they lend themselves to a Lagrangian description in terms of a WZW model and their chiral energy momentum tensor is the one obtained naturally from the usual Sugawara construction. In view of their intricate representation theory they provide interesting examples of conformal field theories., 13 pages, v2: Corrected several typos and minor mistakes, added a reference and streamlined some discussions

Drinfeld Yangian of a queer Lie superalgebra sq 1 is defined as the quantization of a Lie bisuperelgebra of twisted polynomial currents. An analogue of the new system of generators of Drinfeld is being constructed. It is proved for the case Lie superalgebra sq 1 that this so defined Yangian and the Yangian, introduced earlier by M. Nazarov using the FaddeevReshetikhin-Takhtadzhjan approach, are isomorphic. Mathematical Subject Classification (2000). Primary 17B37; Secondary 81R50, 13F60.

In a previous paper the generator matrix elements and (dual) vector reduced Wigner coefficients (RWCs) were evaluated via the polynomial identities satisfied by a certain matrix constructed from the $R$-matrix $R$ and its twisted counterpart $R^T=T\circ R$. Here we provide an alternative evaluation utilising the $R$-matrix $\tilde{R} = (R^T)^{-1}$. This provides a new direct derivation of the vector RWCs obtained indirectly in earlier work via a symmetry relation. This approach has the advantage that it generalises to the Lie superalgebra case, which will be investigated elsewhere., 52 pages

The algebraic structure generated by the creation and annihilation operators of a system of m parafermions and n parabosons, satisfying the mutual parafermion relations, is known to be the Lie superalgebra osp(2m + 1 vertical bar 2n). The Fock spaces of such systems are then certain lowest weight representations of osp(2m + 1 vertical bar 2n). In the current paper, we investigate what happens when the number of parafermions and parabosons becomes infinite. In order to analyze the algebraic structure, and the Fock spaces, we first need to develop a new matrix form for the Lie superalgebra B(n, n) = osp(2n + 1 vertical bar 2n), and construct a new Gelfand-Zetlin basis of the Fock spaces in the finite rank case. The new structures are appropriate for the situation n -> infinity. The algebra generated by the infinite number of creation and annihilation operators is B(infinity, infinity), a well defined infinite rank version of the orthosymplectic Lie superalgebra. The Fock spaces are lowest weight representations of B(infinity, infinity), with a basis consisting of particular row-stable Gelfand-Zetlin patterns.

We describe the finite-dimensional irreducible representations of the Yangian of a Lie superalgebra of type . We formulate and prove a criterion for an irreducible representation to be finite-dimensional.

We construct nonlinear super integrable couplings of the super integrable Dirac hierarchy based on an enlarged matrix Lie superalgebra. Then its super Hamiltonian structure is furnished by super trace identity. As its reduction, we gain the nonlinear integrable couplings of the classical integrable Dirac hierarchy.

In this paper, we give the explicit expressions of the differential operator representations of the exceptional Lie superalgebra D(2, 1;�). Based on these expressions, we construct free field realizations of the currents associated with D(2, 1;�) at an arbitrary level k.

A new eight-dimensional Lie superalgebra is constructed and two isospectral problems with six potentials are designed. Corresponding hierarchies of nonlinear evolution equations, as well as super-AKNS and super-Levi, are derived. Their super-Hamiltonian structures are established by making use of the supertrace identity, and they are integrable in the sense of Liouville.

We study the absolute basis problem in algebras of holomorphic functions in non-commuting variables generating a finite-dimensional nilpotent Lie algebra . This is motivated by J. L. Taylor's programme of non-commutative holomorphic functional calculus in the Lie algebra framework.

Explicit exact solution of supersymmetric Toda fields associated with the Lie superalgebra sl(2|1) is constructed. The approach used is a super extension of Leznov–Saveliev algebraic analysis, which is based on a pair of chiral and antichiral Drienfeld–Sokolov systems. Though such approach is well understood for Toda field theories associated with ordinary Lie algebras, its super analogue was only successful in the super Liouville case with the underlying Lie superalgebra osp(1|2). The problem lies in that a key step in the construction makes use of the tensor product decomposition of the highest weight representations of the underlying Lie superalgebra, which is not clear until recently. So our construction made in this paper presents a first explicit example of Leznov–Saveliev analysis for super Toda systems associated with underlying Lie superalgebras of the rank higher than 1.

A new higher-dimensional Lie algebra is constructed, which is used to generate multiple integrable couplings simultaneously. From this, we come to a general approach for seeking multi-integrable couplings of the known integrable soliton equations.

We study numerical characteristics of varieties of Lie superalgebras and, in particular, the growth of codimensions. An example of an insoluble variety of almost polynomial growth is constructed. We prove that the exponent of this variety is equal to three and calculate the growth exponents for two earlier known soluble varieties.

We construct the Killing superalgebra of supersymmetric backgrounds of ten-dimensional heterotic and type II supergravities and prove that it is a Lie superalgebra. We also show that if the fraction of supersymmetry preserved by the background is greater than 1/2, in the heterotic case, or greater than 3/4 in the type II case, then the background is locally homogeneous.

In a Wigner quantum mechanical model, with a solution in terms of the Lie superalgebra , one is faced with determining the eigenvalues and eigenvectors for an arbitrary self-adjoint odd element of in any unitary irreducible representation W. We show that the eigenvalue problem can be solved by the decomposition of W with respect to the branching . The eigenvector problem is much harder, since the Gel'fand?Zetlin basis of W is involved, and the explicit actions of generators on this basis are fairly complicated. Using properties of the Gel'fand?Zetlin basis, we manage to present a solution for this problem as well. Our solution is illustrated for two special classes of unitary representations: the so-called Fock representations and the ladder representations.

The upper triangular matrix of Lie algebra is used to construct integrable couplings of discrete solition equations. Correspondingly, a feasible way to construct integrable couplings is presented. A nonlinear lattice soliton equation spectral problem is obtained and leads to a novel hierarchy of the nonlinear lattice equation hierarchy. It indicates that the study of integrable couplings using upper triangular matrix of Lie algebra is an important step towards constructing integrable systems.

In this paper N = 4 supersymmetry of generalized Morse oscillators in one dimension is studied. Both bound states and scattering states of its four superpartner Hamiltonians are analyzed by using unitary irreducible representations of the noncompact Lie algebra su(1,1). The spectrum-generating algebra governing the Hamiltonian of the N = 4 supersymmetric Morse oscillator is shown to be connected with the realization of Lie superalgebra osp(1,2) or B(0,1) in terms of the variables of a supersymmetric two-dimensional harmonic oscillator.

Let be the Lie superalgebra . We show how to associate a weight Λ to a composite partition with composite Young diagram . Based upon the definition of critical representations, the notion of 'critical composite partition' is introduced. It is shown that for critical composite partitions (subject to a technical restriction) the corresponding representation VΛ is tame, so its character formula can be computed. This character is shown to coincide with the composite S-function .

We showthat the Casimir operators of the semidirect productsG2−→⊕2�(a,b)⊕�(0,0)h of the exceptional Lie algebraG2 and a Heisenberg algebra h can be constructed explicitly from the Casimir operators of G2.

Triangular deformation of the orthosymplectic Lie superalgebra osp(1|4) is defined by chains of twists. Corresponding classical r-matrix is obtained by a contraction procedure from the trigonometric r-matrix. The carrier space of the constant r-matrix is the Borel subalgebra.

In this work, we present a vertex–face correspondence between an elliptic R-operator and Boltzmann weights related to the Lie superalgebra sl(mn).

Using functional equations we solve the Maurer–Cartan equations and a special version of the classical dynamical Yang–Baxter equation (vCDYBE). Our solutions are valid for any Lie algebra over a base ring containing , and in the case of vCDYBE for any quadratic Lie algebra. Our method applies also to Lie superalgebras.

This is a reply to the comment by P Schlottmann and A A Zvyagin. PACS numbers: 71.10.Fd, 71.27, 75.10.Jm In their comment [1], Schlottmann and Zvyagin raise several issues regarding the nature of integrable impurities in one-dimensional quantum lattice models, and claim to expose false statements in our recent paper [2]. In order to address these issues in a pedagogical manner, we feel that it is appropriate to discuss these questions in terms of models based on the gl(2|1) invariant solution of the Yang–Baxter equation. However, it is important from the outset to make it clear that the arguments we present below are general and apply to other classes of models. The first point that we would like to make is that it is claimed in [1] that there are two approaches to the algebraic Bethe ansatz. However, it appears to us that approach (i) described in point (2) of [1] is the coordinate Bethe ansatz and we do not understand why this should be referred to as an algebraic Bethe ansatz. In the coordinate Bethe ansatz approach, one starts with a prescribed Hamiltonian and then solves the Schrodinger equation to obtain the two-particle scattering matrices and the particle–impurity scattering matrix. Together these scattering matrices form the monodromy matrix. It is impossible to infer the Hamiltonian from such a monodromy matrix. In this context we do not feel that point (1) of [1] answers our query about the existence of the impurity monodromy matrix in the algebraic approach (ii) of point (2) in [1]. Hereafter, we focus our attention on this case. The solution of the Yang–Baxter equation R12(u− v)R13(u)R23(v) = R23(v)R13(u)R12(u− v) (1) associated with the Lie superalgebra gl(2|1) is an operator R(u) ∈ End (V ⊗ V ), where V is a three-dimensional Z2-graded space with one bosonic and two fermionic degrees of freedom. Explicitly, this operator takes the form R(u) = u · I ⊗ I + P (2) where P is the Z2-graded permutation operator. For the purposes of constructing integrable one-dimensional quantum systems on a closed lattice, it is usual to introduce the Yang–Baxter 0305-4470/02/296197+05$30.00 © 2002 IOP Publishing Ltd Printed in the UK 6197

We use Lie algebraic techniques to obtain exact solutions of the effective mass Schrodinger equation. In particular we use the su(1,1) algebra, both as a spectrum generating algebra and as a potential algebra, to obtain exact solutions of effective mass Schrodinger equations corresponding to a number of potentials. We also discuss the construction of isospectral Hamiltonians for which both the mass and the potential are different.

We show a local duality between the left and right finite-dimensional representations on the enveloping algebra of (2)q, the generalized Lie algebra introduced by Lyubashenko and Sudbery. This duality works in a general context of algebras satisfying certain algebraic properties; our goal in this paper is to prove that the enveloping algebra of (2)q satisfies these abstract properties.

In this paper we study certain integrability properties of the supersymmetric sine-Gordon equation. We construct Lax pairs with their zero-curvature representations which are equivalent to the supersymmetric sine-Gordon equation. From the fermionic linear spectral problem, we derive coupled sets of super Riccati equations and the auto-B\"acklund transformation of the supersymmetric sine-Gordon equation. In addition, a detailed description of the associated Darboux transformation is presented and non-trivial super multisoliton solutions are constructed. These integrability properties allow us to provide new explicit geometric characterizations of the bosonic supersymmetric version of the Sym--Tafel formula for the immersion of surfaces in a Lie superalgebra. These characterizations are expressed only in terms of the independent bosonic and fermionic variables.

A new kind of graded Lie algebra (We call it graded Lie algebra) is introduced as a framework for formulating parasupersymmetric theories. By choosing suitable Bose subspace of the graded Lie algebra and using relevant generalized Jacobi identities, we generate the whole algebraic structure of parastatistics.

We show that the (1+1)-dimensional quantum superplane introduced by Manin is a quantum supergroup, according to the Faddeev-Reshetikhin-Takhtajan approach, when it is extended by the inverse of the bosonic variable. We then give its supermatrix element, its corresponding R-matrix and its Hopf structure. This new point of view allows us, first, to realize its dual Hopf superalgebra starting from postulated initial pairings. Second, we construct a right-invariant differential calculus on it and then deduce the corresponding quantum Lie superalgebra which as a commutation superalgebra appears classical, and as Hopf structure is a non-cocommutative q-deformed one.

Based on the linear quantum transformation theory, we further propose a supersymmetric unitary transformation, whose operator can be constructed by the supersymmetric generators of Lie superalgebra. Therefore, by making use of the supersymmetric unitary transformation, we can obtain the eigenspectrum, the eigenstates and the super-partition function for the Super-Jaynes-Cummings model when the coupling constants are Grassmann valued.

The structure of the quantized enveloping algebra of the Lie superalgebra sl(m,1) is studied, crystal bases for its negative part are constructed.

An integrable vertex model associated to the Lie superalgebra sl(2|1) is constructed for the description of the spin quantum Hall critical phase. The model involves R-matrix solutions of the Yang-Baxter equation with respect to both the vector representation of sl(2|1) and its dual and an inhomogenity in the spectral parameters. On the torus the model can be mapped onto a Chalker-Coddington-type network model.

We study the relationship between the equations of first order Lagrangian field theory on fiber bundles and the covariant Hamilton equations on the finite-dimensional polysymplectic phase space of covariant Hamiltonian field theory. The main peculiarity of these Hamilton equations lies in the fact that, for degenerate systems, they contain additional gauge fixing conditions. We develop the BRST extension of the covariant Hamiltonian formalism, characterized by a Lie superalgebra of BRST and anti-BRST symmetries.

The graded contractions of pseudo-Euclidean Lie algebra are studied. The non-equivalent gradings of of type and a are extended to the entire Lie algebra , using the action of on the Abelian ideal (the translations.). The graded contractions embed into a large family of six-dimensional Lie algebras. The family includes solvable, nilpotent and nonsolvable Lie algebras, both decomposable and indecomposable ones. The distinction between graded and Inonu-Wigner contractions is analysed. The physically most interesting Lie algebras obtained by the contractions are the inhomogeneous Galilei and pseudo-Galilei.

A full set of Casimir operators for the Lie superalgebra $gl(m/\infty)$ is constructed and shown to be well defined in the category $O_{FS}$ generated by the highest weight irreducible representations with only a finite number of non-zero weight components. The eigenvalues of these Casimir operators are determined explicitly in terms of the highest weight. Characteristic identities satisfied by certain (infinite) matrices with entries from $gl(m/\infty)$ are also determined.

We present details of a geometric method to associate a Lie superalgebra with a large class of bosonic supergravity vacua of the type AdS x X, corresponding to elementary branes in M-theory and type II string theory., Comment: 16 pages

Firstly we expand a finite-dimensional Lie algebra into a higher-dimensional 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.

As is well known, the type 1 Lie superalgebra admits a one-parameter family of finite-dimensional irreducible representations. We have carried out an analytic Bethe ansatz related to this family of representations. We present formulae, which are deformations of previously proposed determinant formulae labelled by a Young superdiagram. These formulae will provide a transfer matrix eigenvalue in a dressed vacuum form related to the solutions of a graded Yang-Baxter equation, which depend not only on the spectral parameter but also on a non-additive continuous parameter. A class of transfer matrix functional relations among these formulae is briefly mentioned.

The Drinfeld twist is applyed to deforme the rank one orthosymplectic Lie superalgebra $osp(1|2)$. The twist element is the same as for the $sl(2)$ Lie algebra due to the embedding of the $sl(2)$ into the superalgebra $osp(1|2)$. The R-matrix has the direct sum structure in the irreducible representations of $osp(1|2)$. The dual quantum group is defined using the FRT-formalism. It includes the Jordanian quantum group $SL_\xi(2)$ as subalgebra and Grassmann generators as well., Comment: LaTeX, 9 pages

By making use of the vector product in R3, a commuting operation is introduced so that R3 becomes a Lie algebra. The resulting loop algebra 3 is presented, from which the well-known AKNS hierarchy is produced. Again via applying the superposition of the commuting operations of the Lie algebra, a commuting operation in R6 is constructed so that R6 becomes a Lie algebra. Thanks to the corresponding loop algebra 3 of the Lie algebra R3, the integrable coupling of the AKNS system is obtained. The method presented in this paper is rather simple and can be used to work out integrable coupling systems of the other known integrable hierarchies of soliton equations.

The known Holstein-Primakoff and Dyson realizations for the Lie algebras $gl(n+1),\;n=1,2,\ldots$ in terms of Bose operators are generalized to the class of the Lie superalgebras $gl(m/n+1)$ for any $n$ and $m$. Formally the expressions are the same as for $gl(m+n+1)$, however both Bose and Fermi operators are involved., TeX, 7 pages

From the point of view of the Young superdiagram method, an analytic Bethe ansatz is carried out for Lie super algebra . For the transfer matrix eigenvalue formulae in dressed-vacuum form, we present some expressions, which are quantum analogues of Jacobi - Trudi and Giambelli formulae for Lie superalgebra . We also propose transfer-matrix functional relations, which are Hirota bilinear difference equations with some constraints.

Quantum groups at roots of unity have the property that their centre is enlarged. Polynomial equations relate the standard deformed Casimir operators and the new central elements. These relations are important from a physical point of view since they correspond to relations among quantum expectation values of observables that have to be satisfied on all physical states. In this paper, we establish these relations in the case of the quantum Lie superalgebra U_q(sl(2|1)). In the course of the argument, we find and use a set of representations such that any relation satisfied on all the representations of the set is true in U_q(sl(2|1)). This set is a subset of the set of all the finite dimensional irreducible representations of U_q(sl(2|1)), that we classify and describe explicitly.

We give a realization of the quantum affine Lie superalgebras U_q(A(M,N))^(1) in terms of anyons defined on a one or two-dimensional lattice, the deformation parameter q being related to the statistical parameter $\nu$ of the anyons by q = exp(i\pi\nu). The construction uses anyons contructed from usual fermionic oscillators and deformed bosonic oscillators. As a byproduct, realization deformed in any sector of the quantum superalgebras U_q(A(M,N)) is obtained.

An alternative description of the Lie superalgebra osp(2n + 1/2m) in terms of generators and relations is given. The generators, called Green generators, are the root vectors of osp(2n + 1/2m), corresponding to the orthogonal roots.

We consider geometric and supergravity Killing spinors and the spinor bilinears constructed out of them. The spinor bilinears of geometric Killing spinors correspond to the antisymmetric generalizations of Killing vector fields which are called Killing-Yano forms. They constitute a Lie superalgebra structure in constant curvature spacetimes. We show that the Dirac currents of geometric Killing spinors satisfy a Lie algebra structure up to a condition on 2-form spinor bilinears. We propose that the spinor bilinears of supergravity Killing spinors give way to different generalizations of Killing vector fields to higher degree forms. It is also shown that those supergravity Killing forms constitute a Lie algebra structure in six and ten dimensional cases. For five and eleven dimensional cases, the Lie algebra structure depends on an extra condition on supergravity Killing forms., Comment: 17 pages, published version

We recall the relation between the Lie superalgebra $osp(1/2n)$ and para-Bose operators. The quantum superalgebra $U_q[osp(1/2n)]$, defined as usual in terms of its Chevalley generators, is shown to be isomorphic to an associative algebra generated by so-called pre-oscillator operators satisfying a number of relations. From these relations, and the analogue with the non-deformed case, one can interpret these pre-oscillator operators as deformed para-Bose operators. Some consequences for $U_q[osp(1/2n)]$ (Cartan-Weyl basis, Poincar\'e-Birkhoff-Witt basis) and its Hopf subalgebra $U_q[gl(n)]$ are pointed out. Finally, using a realization in terms of ``$q$-commuting'' $q$-bosons, we construct an irreducible finite-dimensional unitary Fock representation of $U_q[osp(1/2n)]$ and its decomposition in terms of $U_q[gl(n)]$ representations when $q$ is a root of unity., Comment: 15 pages, LaTeX (latex twice), no figures