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In this paper, we define generalized Casimir operators for a loop contragredient Lie superalgebra and prove that they commute with the underlying Lie superalgebra. These operators have applications in the decomposition of tensor product modules. We further introduce the notion of generalized Gelfand invariants for the loop general linear Lie superalgebra and show that they also commute with the underlying Lie superalgebra. These operators when applied to a highest weight vector in a tensor product module again induce a new highest weight vector.

New convolution identities for orthogonal polynomials belonging to the $q=-1$ analog of the Askey-scheme are obtained. A specialization of the Chihara polynomials will play a central role as the eigenfunctions of a special element of the Lie superalgebra $\mathfrak{osp}(1|2)$ in the positive discrete series representation. Using the Clebsch-Gordan coefficients, a convolution identity for the Specialized Chihara, the dual -1 Hahn and the Big -1 Jacobi polynomials is found. Using the Racah coefficients, a convolution identity for the Big -1 Jacobi and the Bannai-Ito polynomials is found. Finally, these results are applied to construct a bilinear generating function for the Big -1 Jacobi polynomials., Comment: 19 pages

Based on the Lie superalgebra spl(2, 1) and the TAH scheme, we generate (1+1)-dimensional and (2+1)-dimensional super soliton hierarchies associated with spl(2, 1). And by using the super-trace identity on Lie superalgebras, we construct the super-Hamiltonian structures of these new super soliton hierarchies. At last, we also obtain the Darboux transformations of the (1+1)-dimensional super soliton hierarchy.

In this paper we study the finite W-algebra for the queer Lie superalgebra Q(n) associated with the non-regular even nilpotent coadjoint orbits in the case when the corresponding nilpotent element has Jordan blocks each of size l. We prove that this finite W-algebra is isomorphic to a quotient of the super-Yangian of Q({n/l}), arXiv admin note: text overlap with arXiv:1403.3866

We formulate the algebraic version of recoupling theory suitable for commutation quantization over any gradation. This gives a generalization of graded Lie algebra. Underlying this is the new notion of an ω-algebra defined in this paper. ω-algebra is a generalization of algebra that goes beyond nonassociativity. We construct the universal enveloping ω-algebra of recoupling Lie algebras and prove a generalized Poincare–Birkhoff–Witt theorem. As an example we consider the algebras over an arbitrary recoupling of Zn graded Heisenberg Lie algebra. Finally we uncover the usual coalgebra structure of a universal envelope and substantiate its Hopf structure.

A grading of a Lie algebra is called fine if it cannot be further refined. Fine gradings provide basic information about the structure of the algebra. There are six fine gradings of the semisimple Lie algebra of type A1×A1 over the complex number field. An explicit description of all the fine gradings of A1×A1 is given in terms of the four-dimensional representation o(4,C) of the algebra.

A grading of a Lie algebra is called fine if it cannot be further refined. Fine gradings provide basic information about the structure of the algebra. There are eight fine gradings of the simple Lie algebra of type A3 over the complex number field. One of them (root decomposition) is the main tool of the theory and applications in working with A3 and with its representations; one other has also been used in the literature, and the rest have apparently not been recognized so far. An explicit description of all the fine gradings of A3 is given in terms of the four-dimensional [sl(4, C)] and six-dimensional orthogonal [o(6, C)] representations of the algebra. These results should be useful generally for choosing bases which reflect structural properties of the Lie algebra, for defining various sets of additive quantum numbers for systems with such symmetries, and for systematic study of grading preserving contractions of this Lie algebra.

This paper deals with a systematic construction of higher spin operators, defined as conformally invariant differential operators acting on functions on flat space R-m with values in an arbitrary half-integer irreducible representation for the spin group. To be more precise, the higher spin version of the Dirac operator and associated twistor operators will be constructed as generators of a transvector algebra, hereby generalising the well-known fact that the classical Dirac operator on Rm and its symbol generate the orthosymplectic Lie superalgebra osp(1, 2). To do so, we will use the extremal projection operator and its relation to transvector algebras. In the second part of the article, the conformal invariance of the constructed higher spin operators will be proven explicitly. (C) 2014 AIP Publishing LLC.

We will consider a supergeneralization of the Duffin–Kemmer–Petiau algebra. It has been shown to be intimately related to the Lie super algebra osp(N,M). We will also discuss its representations.

There are three fine gradings of the simple Lie algebra of type B2 over the complex number field. They provide a basic information about the structure of the algebra. In the paper an explicit description of all fine gradings is given in terms of the four-dimensional symplectic [sp(4, C)] and five-dimensional orthogonal [o(5, C)] representations of the algebra. In addition, the real forms of B2 are considered. It is shown which of the fine gradings survive the restriction to each of the real forms. These results should be useful in defining various sets of additive quantum numbers for systems with such symmetries, for systematic study of grading preserving contractions of this Lie algebra, and generally for choosing bases which reflect structural properties of the Lie algebra.

A graded minimal Lie algebraic extension of the space–time symmetry is constructed involving only spin-1 multiplets as novel generators. The extension involves Z4×Z4 graded parameters and generators. It provides a bosonic analog to supersymmetry since the composition of three symmetric vector charges produces a space–time translation. There arise three noncommutative four-dimensional manifolds with pseudometric.

We give a new derivation of the Manakov and the product case integrability conditions of the Euler equations on Lie algebra so(4). Fourth first integral functionally independent of the three already known integrals is obtained explicitly for all values of the parameters by an algorithmic approach.

The structure and representations of the quantum supergroup OSPq(2|2n) are studied systematically. The algebra of functions on the quantum supergroup, which specifies the quantum supergroup itself, is taken to be the superalgebra generated by the matrix elements of the vector representation of the quantized universal superalgebra Uq(osp(2|2n)). It is shown that the algebra of functions is dense in the full dual Uq(osp(2|2n))* of Uq(osp(2|2n)) and possesses a Hopf superalgebra structure. The left integral and right integral on the quantum supergroup are discussed. Induced representations are developed using the noncommutative geometry of quantum homogeneous supervector bundles, and a geometric realization of irreducible representations is obtained.

We construct a Kac–Moody superalgebra L as the minimal graded Lie superalgebra with local part V*⊕ge⊕V, where g is a “smaller” Lie superalgebra inside L, V is an irreducible highest weight g-module, and V* is the contragredient of V. We show that the weight multiplicities of irreducible highest weight modules over Kac–Moody superalgebras of finite type and affine type [more precisely, Kac–Moody superalgebras of type B(0,r), B(1)(0,r), A(4)(2r,0), A(2)(2r−1,0), and C(2)(r+1)] are given by polynomials in the rank r. The degree of these weight multiplicity polynomials are less than or equal to the depth of weights.

The co-Lie structures compatible with the osp(2/2) Lie superalgebra structure are investigated and found to be all of the coboundary type. The corresponding classical r-matrices are classified into several disjoint families. The osp(1/2)⊕u(1) Lie superbialgebras are also classified.

It is shown that the quantized enveloping algebra of sl(n) contains a generalized Lie algebra, defined by means of axioms similar to Woronowicz’s. This gives rise to Lie algebra-like generators and relations for the locally finite part of the quantized enveloping algebra, and suggests a canonical Poincare–Birkhoff–Witt basis.

Complete solution of the decomposition problem for finite-dimensional representations π of the BRST algebra A is presented. As shown earlier by the authors [Rev. Math. Phys. 5, 191 (1993)], A coincides with the Lie superalgebra l(1,1). It is proved that an arbitrary π either has the decomposition into a direct sum of irreducible and/or indecomposable representations (IR and IDR, respectively) or has the set of indecomposable subrepresentations that does not admit any classification. All the series of IDR are explicitly described, and all the unclassifiable cases are reduced to definite unsolvable algebraic problems. The absence of classification is established by means of a special computer method, which opens a new possibility to search for unsolvable algebraic problems.

We apply the method of the central extensions introduced by Reshetikhin and Semenov–Tian–Shansky to the case of the Perk–Shultz model. By using the method proposed by Frenkel–Ding, we establish the Drinfeld constructions of the quantum affine superalgebra Uq(gl(m/n)).

Supersymmetric quantum mechanics cannot be deformed when the superposition of only a pair of usual bosons and fermions is considered, but it can if nontrivial parabosons and parafermions of the same order p of paraquantization are superposed. We take the simplest case p=2 and exhibit reducibility problems in that context by using Green–Cusson Ansatze following Macfarlane methods. Specific representations of the Lie superalgebra osp (2|2, R) play an interesting role in connection with possible deformations.

We prove that the second cohomology group with coefficients in the adjoint module of both the infinite‐dimensional Lie superalgebras k(1) and k+(1), as well as their unique central extensions, which are known as the Neveu–Schwarz and the Ramond superalgebras, respectively, is equal to zero. This particularly implies that all these Lie superalgebras are rigid, i.e. they can only be deformed in a trivial manner.

In a paper by C. N. Pope and K. S. Stelle [Phys. Lett. B 226, 257–263 (1989)], it was shown that the algebra of local area‐preserving diffeomorphisms on the torus is equivalent to su(∞). In this paper we present a few remarks on the universal central extension of this infinite dimensional Lie algebra, and construct, for this case, in analogy to the situation in which the affine Lie algebra has an underlying finite‐dimensional Lie algebra, the nontrivial universal central extension.

It is shown that the orthosymplectic Lie superalgebras osp (m/2n) have a nonregular subalgebra osp(1/2n). This implies that paraboson operators can be realized as elements of osp(m/2n). The embedding osp(2/2n)⊇osp(1/2n), or, in a different notation, C(n+1)⊇B(0,n), is studied in more detail. In particular, branching rules are determined for all typical and atypical irreducible representations of osp(2/2n) with respect to the subalgebra osp(1/2n). Finally, dimension and superdimension formulas are given for the Lie superalgebras under consideration.

Lie superalgebraic methods are used to establish a connection between the huge Lie superalgebra Ξ of super‐ (pseudo‐) differential operators and various super KP‐hierarchies. We show in particular that Ξ splits into 5=2×2+1 graded algebras expected to correspond to five classes of super‐KP‐hierarchies generalizing the well‐known Manin–Radul and Figueroa–Mas–Ramos supersymmetric KP‐hierarchies.

We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of $U(g)$. The construction of such differential structures is interpreted in terms of colour Lie superalgebras.

The construction approach proposed in the previous paper Ref. 1 allows us there and in the present paper to construct at generic deformation parameter $q$ all finite--dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$. The finite--dimensional $U_{q}[gl(2/2)]$-modules $W^{q}$ constructed in Ref. 1 are either irreducible or indecomposible. If a module $W^{q}$ is indecomposible, i.e. when the condition (4.41) in Ref. 1 does not hold, there exists an invariant maximal submodule of $W^{q}$, to say $I_{k}^{q}$, such that the factor-representation in the factor-module $W^{q}/I_{k}^{q}$ is irreducible and called nontypical. Here, in this paper, indecomposible representations and nontypical finite--dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$ are considered and classified as their module structures are analized and the matrix elements of all nontypical representations are written down explicitly., Comment: Latex file, 49 pages

Different statistics generalizing the Fermi–Dirac one, i.e., statistics associated with parafermions of order p, orthofermions, deformed parafermions, and genons, are studied in connection with Lie superalgebras. The class spl(m/n) appears as the fundamental one in this context. A specific relation between parafermions and orthofermions is also pointed out.

We present in detail the classification of the finite dimensional irreducible representations of the super Yangian associated with the Lie superalgebra $gl(1|1)$., 14 pages, plain latex, no figure

We study finite W-algebras associated to even regular (principal) nilpotent elements for the family of simple exceptional Lie superalgebras D(2, 1; α) and for the universal central extension of 𝔭𝔰𝔩(2|2). We give a complete description of these finite W-algebras in terms of generators and relations.

It is shown that the second order Casimir operators C and Cq of the Lie superalgebra osp(1‖2) and the quantized Lie superalgebra ospq(1‖2), respectively, possess natural square roots.

Although character formulas for simple finite‐dimensional modules of the Lie superalgebra sl(m/n) are not known in general, they are known in the cases of so‐called typical and of singly atypical modules. For singly atypical modules, however, the character formula is such that the classical limiting procedure yielding a dimension formula from a character formula does, in general, not work. In this article, another approach is presented yielding an explicit dimension formula for all singly atypical modules of sl(m/n) in terms of the highest weight labels.

G‐graded contractions of the complex Lie algebra B2 and of its real forms are considered. It is found that a particular G=Z2×Z2 is a proper choice of the grading group preserved when graded contractions leading to Lie algebras of the kinematical groups of space–time are studied.

A deformed universal enveloping superalgebra Uq of the Lie superalgebra osp(3/2), which is a Hopf algebra, is defined as freely generated by deformed para‐Bose operators B± and deformed para‐Fermi operators F±. The q analog of the Cartan–Weyl basis and all deformed supercommutation relations between the generators are written down. A morphism of Uq onto a deformed Clifford–Weyl superalgebra Wq(1/1)a is constructed, where Wq(1/1)a is generated by q bosons b± and q fermions f± under the condition that the q bosons anticommute with the q fermions. The corresponding oscillator representation is studied.

We construct the $N=2$ super $W_4$ algebra as a certain reduction of the second Gel'fand-Dikii bracket on the dual of the Lie superalgebra of $N=1$ super pseudo-differential operators. The algebra is put in manifestly $N=2$ supersymmetric form in terms of three $N=2$ superfields $\Phi_i(X)$, with $\Phi_1$ being the $N=2$ energy momentum tensor and $\Phi_2$ and $\Phi_3$ being conformal spin $2$ and $3$ superfields respectively. A search for integrable hierarchies of the generalized KdV variety with this algebra as Hamiltonian structure gives three solutions, exactly the same number as for the $W_2$ (super KdV) and $W_3$ (super Boussinesq) cases.

It is shown that the naive Serre presentation corresponding to the simplest Cartan matrix of the special linear Lie superalgebra sl(m,n) yields, for m,n≥2, a larger Lie superalgebra s(m,n) of which sl(m,n) is a nontrivial quotient. The missing relations are found and the q deformation of the universal enveloping algebra of sl(m,n) is constructed. Some auxiliary results related to associative algebras and e Lie algebras defined in terms of generators and relations are also included.

It is shown how using the graded Faddeev–Reshetikhin–Takhtajan (FRT) formalism, one can multiparametrize the quantum superalgebra slq(n‖m). Since the naive Serre–Chevalley relations are not adequate for the presentation of this algebra, the quantization is carried out in the Cartan–Weyl basis. The final results can be generalized to any finite dimensional superalgebra with a symmetrizable Cartan matrix.

A reducible representation of the simple Lie superalgebra osp(1,2) is constructed from two pairs of boson and one pair of fermion creation and annihilation operators. The representation contains in direct sum every distinct (up to equivalence), finite‐dimensional, irreducible representation of osp(1,2) exactly once and so defines a model, generalizing to osp(1,2) Schwinger’s boson model of su(2). The model of osp(1,2) is further generalized to a model of the quantum superalgebra Uq[osp(1,2)] in the case that q is not a root of unity. Matrix elements of generators and basic properties of the R matrix, known from previous studies, are rederived using the boson–fermion calculus.

A {1+1/2l(l−1)}‐parameter deformation of the universal enveloping algebra of the simple Lie algebra Al(=sl(l+1, C)) is derived for all l≥2, starting from a certain multiparameter matrix that satisfies the quantum Yang–Baxter equation. It is shown that this Hopf algebra has the same product relations and antipode as the standard one‐parameter deformation Uq(sl(l+1,C)), but has a different coproduct. It is also shown that for all l≥2 there exists a 1/2l(l−1)‐parameter Hopf algebra whose product relations are merely the commutation relations of Al itself, but whose coproduct is different from the usual one for the universal enveloping algebra of Al.

In this paper, some simple modules of the affine Lie superalgebra A(1)(1,0) are considered which, although they are neither highest nor lowest weight modules, are constructed in a manner closely analogous to the construction of highest weight modules of the finite‐dimensional Lie superalgebras A(m,n).

The modules of the orthosymplectic Lie superalgebra osp(3/2), induced from finite‐dimensional irreducible submodules of the stability subalgebra so(3)⊕gl(1) are investigated. The corresponding infinite‐dimensional irreducible or indecomposable modules, the Kac modules, and the related typical and atypical modules are studied in detail. Every such module is decomposed into a direct sum of either indecomposable or irreducible modules of the even subalgebra so(3)⊕sp(2). For each of these (infinite‐dimensional or finite‐dimensional, irreducible or indecomposable) modules relations are written down, giving the transformations of the basis under the action of the algebra generators.

The commutation relation for coupled SO(3) tensor operators is constructed. This gives rise to a Lie algebra for which 9j symbols appear in the structure constants. In an example it is shown how an exceptional Lie subalgebra can be realized. The closure requirement for the exceptional Lie algebra does not seem to be related to nontrivial 9j symbols but rather to vanishings of relations involving 6j symbols.

A general method is developed for constructing invariants of quantum supergroups using universal R matrices. Applied to Uq(gl(2/1)), this method yields the invariants of this quantum supergroup in explicit form. The eigenvalues of these invariants in arbitrary irreducible highest weight representations are computed.

Using an extension P(n) of the noncontragredient superalgebra P(n), on which a nondegenerate Killing form can be constructed, simple root systems (SRSs) are introduced. To each SRS is associated a Dynkin‐like diagram. Then the maximal regular sub‐(super)algebras are obtained. Moreover, a procedure to construct highest weight irreducible representations of P(n) is presented.

New properties on parabosonic and parafermionic Cusson’s realizations of the Green’s Anzatze are first obtained. Then paraproperties are shown to have a specific interest for our real world, where bosons and fermions have to be superposed if Nature is supersymmetric, these results being obtained on supersymmetric harmonic oscillators. In particular, from the Green–Cusson Ansatze, we construct a class of supersymmetric Hamiltonians that are more general than the ones issued from standard or spin–orbit coupling considerations. We explicitly consider the simplest (previously unknown) super‐Hamiltonian constructed from paraproperties in quantum mechanics and establish its invariance Lie superalgebra.

It is shown that from each self-dual representation of a quantum supergroup with nonvanishing q-superdimension, a graded representation of the Temperley-Lieb algebra can be constructed, which, in turn, gives rise to a solution of the graded Yang-Baxter equation. As examples, the vector irrep of Uq (osp(M/2n)) and all the finite-dimensional irreps of Uq (osp(1/2)) are studied, and the corresponding graded Ř-matrices are obtained explicitly. A general method is also given to obtain link polynomials from graded Ř-matrices. Applied to the ones associated with graded representations of the Temperley-Lieb algebra, the method yields the Jones polynomial.

An inhomogeneous differential realization (IHDR) of Lie superalgebras, which plays an important role in the search for quasi‐exactly solvable problems (QESP) of quantum mechanics, is studied. The corresponding inhomogeneous boson–fermion realization (IHBFR) of Lie superalgebras is obtained. By making use of the IHBFR, a new kind of indecomposable and irreducible representations of Lie superalgebras is studied on the universal enveloping algebra of Heisenberg–Weyl superalgebra, and on its subspaces and quotient spaces. The finite dimensional representations of Lie superalgebras are naturally obtained on the subspaces of Fock space. As examples, the IHDR of Lie superalgebra spl(m,1) is given, and the indecomposable and irreducible representations of Lie superalgebra su(2,1) are studied in detail.

Let g be a finite‐dimensional Lie superalgebra over a graded‐commutative local augmented Arens–Michael ground algebra Λ. A Lie group is associated with a number of pleasant properties to the even sector g0 viewed as an ordinary complete locally convex Lie algebra (which is in general infinite dimensional).

The irreducible representations of the Lie superalgebra gl(m‖n) with highest weights of the form Λ=(λ1,λ2,...λm‖ω) are investigated using a recently introduced induced module construction for atypical modules. The gl(m‖n)↓gl(m‖n−1) branching rules are obtained and a suitable Gel’fand–Tsetlin basis is introduced. The class of representations considered includes some multiply atypical irreducible representations of gl(m‖n) and all irreducible representations of gl(m‖1).

The algebraic cohomology and the spectral sequences for a Z2‐graded Lie algebra are briefly reviewed. The reducibility property of a strongly semisimple Lie superalgebra is established. The role of second and third cohomologies in the deformation of a Lie superalgebra is discussed. Using spectral sequences, the second cohomology of the full BRS algebra is shown to be the ground field and the third cohomology being trivial implies that osp(1,2) is the only graded Lie algebra obtained by deformation of the full BRS algebra. A similar analysis yields the superconformal algebra as a deformation of the super Poincare algebra. The superconformal algebra so derived contains so(4,1) as the even part, ruling out the existence of negative curvature of a de Sitter universe!

The Fock representation of the Lie superalgebra osp(∞‖∞), from which is derived the super boson–fermion correspondence of osp(∞‖∞) is discussed. Solutions of the OSp–SKP hierarchy in terms of the neutral super free fermions are constructed.

The two types of * and grade‐* representations of the Lie superalgebra C(n+1) are classified. A type (1) irreducible *‐representation is characterized by the single condition (Λ,αs)≥0, Λ and αs being the highest weight and the odd simple root, respectively, while the type (2) *‐representations are duals of type (1) *‐representations. For n>1, only the identity and vector representations of C(n+1) are shown to be grade ‐*. The superalgebra C(2) proves to be an exception and admits several classes of nontrivial irreducible grade ‐* representations.