Weak Secrecy in the Multi-Way Untrusted Relay Channel with Compute-and-Forward

We investigate the problem of secure communications in a Gaussian multi-way relay channel applying the compute-and-forward scheme using nested lattice codes. All nodes employ half-duplex operation and can exchange confidential messages only via an untrusted relay. The relay is assumed to be honest but curious, i.e., an eavesdropper that conforms to the system rules and applies the intended relaying scheme. We start with the general case of the single-input multiple-output (SIMO) L-user multi-way relay channel and provide an achievable secrecy rate region under a weak secrecy criterion. We show that the securely achievable sum rate is equivalent to the difference between the computation rate and the multiple access channel (MAC) capacity. Particularly, we show that all nodes must encode their messages such that the common computation rate tuple falls outside the MAC capacity region of the relay. We provide results for the single-input single-output (SISO) and the multiple-input single-input (MISO) L-user multi-way relay channel as well as the two-way relay channel. We discuss these results and show the dependency between channel realization and achievable secrecy rate. We further compare our result to available results in the literature for different schemes and show that the proposed scheme operates close to the compute-and-forward rate without secrecy.Comment: submitted to JSAC Special Issue on Fundamental Approaches to Network Coding in Wireless Communication System

Tensor operators and Wigner-Eckart theorem for the quantum superalgebra U_{q}[osp(1\mid 2)]

Tensor operators in graded representations of Z_{2}-graded Hopf algebras are defined and their elementary properties are derived. Wigner-Eckart theorem for irreducible tensor operators for U_{q}[osp(1\mid 2)] is proven. Examples of tensor operators in the irreducible representation space of Hopf algebra U_{q}[osp(1\mid 2)] are considered. The reduced matrix elements for the irreducible tensor operators are calculated. A construction of some elements of the center of U_{q}[osp(1\mid 2)] is given.Comment: 16 pages, Late

Classification of N=6 superconformal theories of ABJM type

Studying the supersymmetry enhancement mechanism of Aharony, Bergman, Jafferis and Maldacena, we find a simple condition on the gauge group generators for the matter fields. We analyze all possible compact Lie groups and their representations. The only allowed gauge groups leading to the manifest N=6 supersymmetry are, up to discrete quotients, SU(n) x U(1), Sp(n) x U(1), SU(n) x SU(n), and SU(n) x SU(m) x U(1) with possibly additional U(1)'s. Matter representations are restricted to be the (bi)fundamentals. As a byproduct we obtain another proof of the complete classification of the three algebras considered by Bagger and Lambert.Comment: 18 page

Cohomology of Lie superalgebras and of their generalizations

The cohomology groups of Lie superalgebras and, more generally, of color Lie algebras, are introduced and investigated. The main emphasis is on the case where the module of coefficients is non-trivial. Two general propositions are proved, which help to calculate the cohomology groups. Several examples are included to show the peculiarities of the super case. For L = sl(1|2), the cohomology groups H^1(L,V) and H^2(L,V), with V a finite-dimensional simple graded L-module, are determined, and the result is used to show that H^2(L,U(L)) (with U(L) the enveloping algebra of L) is trivial. This implies that the superalgebra U(L) does not admit of any non-trivial formal deformations (in the sense of Gerstenhaber). Garland's theory of universal central extensions of Lie algebras is generalized to the case of color Lie algebras.Comment: 50 pages, Latex, no figures. In the revised version the proof of Lemma 5.1 is greatly simplified, some references are added, and a pertinent result on sl(m|1) is announced. To appear in the Journal of Mathematical Physic

General form of the deformation of Poisson superbracket on (2,2)-dimensional superspace

Continuous formal deformations of the Poisson superbracket defined on compactly supported smooth functions on n-dimensional space taking values in a Grassmann algebra with m generating elements are described up to an equivalence transformation for the case n=m=2. It is shown that in this case the Poisson superalgebra has an additional deformation comparing with other superdimensions (n,m).Comment: LaTex, 13 page

Uqosp(2,2)U_q osp(2,2) Lattice Models

In this paper I construct lattice models with an underlying $U_q osp(2,2)$ superalgebra symmetry. I find new solutions to the graded Yang-Baxter equation. These {\it trigonometric} $R$-matrices depend on {\it three} continuous parameters, the spectral parameter, the deformation parameter $q$ and the $U(1)$ parameter, $b$, of the superalgebra. It must be emphasized that the parameter $q$ is generic and the parameter $b$ does not correspond to the `nilpotency' parameter of \cite{gs}. The rational limits are given; they also depend on the $U(1)$ parameter and this dependence cannot be rescaled away. I give the Bethe ansatz solution of the lattice models built from some of these $R$-matrices, while for other matrices, due to the particular nature of the representation theory of $osp(2,2)$, I conjecture the result. The parameter $b$ appears as a continuous generalized spin. Finally I briefly discuss the problem of finding the ground state of these models.Comment: 19 pages, plain LaTeX, no figures. Minor changes (version accepted for publication

A super-analogue of Kontsevich's theorem on graph homology

In this paper we will prove a super-analogue of a well-known result by Kontsevich which states that the homology of a certain complex which is generated by isomorphism classes of oriented graphs can be calculated as the Lie algebra homology of an infinite-dimensional Lie algebra of symplectic vector fields.Comment: 15 page

Comments on Drinfeld Realization of Quantum Affine Superalgebra Uq[gl(m∣n)(1)]U_q[gl(m|n)^{(1)}] and its Hopf Algebra Structure

By generalizing the Reshetikhin and Semenov-Tian-Shansky construction to supersymmetric cases, we obtain Drinfeld current realization for quantum affine superalgebra $U_q[gl(m|n)^{(1)}]$. We find a simple coproduct for the quantum current generators and establish the Hopf algebra structure of this super current algebra.Comment: Some errors and misprints corrected and a remark in section 4 removed. 12 pages, Latex fil

Conserved Charges in the Principal Chiral Model on a Supergroup

The classical principal chiral model in 1+1 dimensions with target space a compact Lie supergroup is investigated. It is shown how to construct a local conserved charge given an invariant tensor of the Lie superalgebra. We calculate the super-Poisson brackets of these currents and argue that they are finitely generated. We show how to derive an infinite number of local charges in involution. We demonstrate that these charges Poisson commute with the non-local charges of the model