Lie Superbialgebras Research Articles

Cosmological string backgrounds from super Poisson-Lie T-plurality

We generalize the formulation of Poisson-Lie (PL) T-plurality proposed by R. von Unge (2002) [6] from Lie groups to Lie supergroups. By taking a convenient ansatz for metric of the σ-model in terms of the left-invariant one-forms of the isometry Lie supergroups (C3+A) and GL(1|1) we construct cosmological string backgrounds, including (2+1|2)-dimensional metric, time-dependent dilaton and vanishing torsion, in a way that they satisfy the one-loop beta-function equations. Starting from the decompositions of semi-Abelian Drinfeld superdoubles (DSDs) generated by the (C3+A) and gl(1|1) Lie super bi-algebras we find the conformal duality/plurality chains of 2+1-dimensional cosmological string backgrounds coupling with two fermionic fields. In particular, the new backgrounds obtained by the super PL T-plurality remain conformally invariant at one-loop level. This work can prompt many new insights into supergravity and obviously has interesting mathematical relations with double field theory.

Lie Super-bialgebra Structures on the Super-BMS3 Algebra

In this paper, Lie super-bialgebra structures on the super-BMS3 algebra are considered. It is proved that all such Lie super-bialgebras are triangular coboundary Lie super-bialgebras. The method we use is mainly based on the computation of derivations from the super-BMS3 to the tensor product of its adjoint module.

Lie superbialgebra structures on the Lie superalgebra C3+A and deformation of related integrable Hamiltonian systems

Admissible structure constants related to the dual Lie superalgebras of particular Lie superalgebra (C3+A) are found by straightforward calculations from the matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation. Then, by making use of the automorphism supergroup of the Lie superalgebra (C3+A), the Lie superbialgebra structures on the Lie superalgebra (C3+A) are obtained and classified into inequivalent 31 families. We also determine all corresponding coboundary and bi-r-matrix Lie superbialgebras. The quantum deformations associated with some Lie superbialgebras (C3+A) are obtained, together with the corresponding deformed Casimir elements. As an application of these quantum deformations, we construct a deformed integrable Hamiltonian system from the representation of the Hopf superalgebra Uλ(Cp=12,𝜖⊕A1,1)(C3+A).

Lie Super-bialgebra Structures on Generalized Ramond N=2 Super-Virasoro Algebras

In this paper, Lie super-bialgebra structures on generalized Ramond N=2 super-Virasoro algebras [Formula: see text] are determined. It is proved that all Lie super-bialgebras are coboundary triangular Lie super-bialgebras.

The [formula omitted] Lie superbialgebras

By direct calculations of matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation, and using the automorphism supergroup of the gl(1|1) Lie superalgebra, we determine and classify all gl(1|1) Lie superbialgebras. Then, by calculating their classical r-matrices, the gl(1|1) coboundary Lie superbialgebras and their types (triangular, quasi-triangular or factorizable) are determined, furthermore in this way super Poisson structures on the GL(1|1) Lie supergroup are obtained. Also, we classify Drinfeld superdoubles based on the gl(1|1) as a theorem. Afterwards, as a physical application of the coboundary Lie superbialgebras, we construct a new integrable system on the homogeneous superspace OSp(1|2)/U(1). Finally, we make use of the Lyakhovsky and Mudrov formalism in order to build up the deformed gl(1|1) Lie superalgebra related to all gl(1|1) coboundary Lie superbialgebras. For one case, the quantization at the supergroup level is also provided, including its quantum R-matrix.

Classical r-matrices of two- and three-dimensional Lie superbialgebras and their Poisson-Lie supergroups

We obtain the classical r-matrices of two and three dimensional Lie super-bialgebras. We thus classify all two and three dimensional coboundary Lie super-bialgebras and their types (triangular, quasi-triangular, or factorable). Using the Sklyanin superbracket, we then obtain the super Poisson structures on the related Poisson-Lie supergroups.

Classification of four and six dimensional Drinfel’d superdoubles

Using adjoint representation we first classify two and three dimensional Lie superbialgebras obtained from decomposable Lie superalgebras. In this way we complete the classification obtained by Eghbali et al., [J. Math. Phys. 51, 073503 (2010); e-print arXiv:0901.4471 [math-ph]]. Then we classify all four and six dimensional Drinfel’d superdoubles.

Classification of two and three dimensional Lie superbialgebras

Using adjoint representation of Lie superalgebras, we obtain the matrix form of super-Jacobi and mixed super-Jacobi identities of Lie superbialgebras. By direct calculations of these identities, and use of automorphism supergroups of two and three dimensional Lie superalgebras, we obtain and classify all two and three dimensional Lie superbialgebras.

Lie super-bialgebra structures on generalized super-Virasoro algebras

In this article, Lie super-bialgebra structures on generalized super-Virasoro algebras ℒ are considered. It is proved that all such Lie super-bialgebras are coboundary triangular Lie super-bialgebras if and only if H 1(ℒ, ℒ ⊗ ℒ)=0.

Poisson-Lie T-dual sigma models on supermanifolds

We investigate Poisson-Lie symmetry for T-dual sigma models on supermanifolds in general and on Lie supergroups in particular. We show that the integrability condition on this super Poisson-Lie symmetry is equivalent to the super Jacobi identities of the Lie super-bialgebras. As examples we consider models related to four dimensional Lie super-bialgebras ((2A1,1+2A)1,D10p = ½) and ((2A1,1+2A)1,I). Then generally it is shown that for Abelian case (g,I) the super Poisson-Lie T-duality transforms the role of fermionic (bosonic) fields in the model to bosonic (fermionic) fields on the dual model and vice versa.

Lie super-bialgebra structures on the Ramond N = 2 super-virasoro algebra

In this paper, Lie super-bialgebra structures on Ramond N = 2 super-Virasoro algebras L are considered. It is proved that all such Lie super-bialgebras are coboundary triangular Lie super-bialgebras.

Etingof–Kazhdan quantization of Lie superbialgebras

For every semi-simple Lie algebra g one can construct the Drinfeld–Jimbo algebra U h DJ ( g ) . This algebra is a deformation Hopf algebra defined by generators and relations. To study the representation theory of U h DJ ( g ) , Drinfeld used the KZ-equations to construct a quasi-Hopf algebra A g . He proved that particular categories of modules over the algebras U h DJ ( g ) and A g are tensor equivalent. Analogous constructions of the algebras U h DJ ( g ) and A g exist in the case when g is a Lie superalgebra of type A-G. However, Drinfeld's proof of the above equivalence of categories does not generalize to Lie superalgebras. In this paper, we will discuss an alternate proof for Lie superalgebras of type A-G. Our proof utilizes the Etingof–Kazhdan quantization of Lie (super)bialgebras. It should be mentioned that the above equivalence is very useful. For example, it has been used in knot theory to relate quantum group invariants and the Kontsevich integral.

Classical r-matrices for the osp(2/2) Lie superalgebra

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.

Classification of low-dimensional Lie super-bialgebras

A thorough analysis of Lie super-bialgebra structures on Lie super-algebras osp(1;2) and super e(2) is presented. A combined technique of computer algebraic computations and subsequent identification of equivalent structures is applied. In all the cases, Poisson–Lie brackets on supergroups are found. The possibility of quantizing them in order to obtain quantum groups is discussed. It turns out to be straightforward for all but one structure for the super-E(2) group.

Duals of quasitriangular Hopf superalgebras and the classical limit

The duality of quasitriangular Hopf superalgebras and superbialgebras with so-called main commutation relations (MCR), which is established by means of bilinear forms with values in a finite-dimensional representation of the universal R-matrix R, is used to calculate relations of quantum supercoordinates for several examples. The tensor product of representations of R, with respect to the comultiplication, then leads to the construction of an accordingly deformed tensor product of MCR-type superbialgebras. The correspondingly deformed tensor product of quantum vector superspaces is then established. In the classical limit of Lie superbialgebras one similarly obtains supercommutation relations, which may be understood, in case of quantum groups with deformation parameter q ≠ 1, as MCR in the first order of ln q. Again the deformed tensor product of such semiclassical MCR-type Lie superbialgebras is constructed.