Quasi-Hopf Superalgebras and Elliptic Quantum Supergroups

We introduce the quasi-Hopf superalgebras which are $Z_2$ graded versions of Drinfeld's quasi-Hopf algebras. We describe the realization of elliptic quantum supergroups as quasi-triangular quasi-Hopf superalgebras obtained from twisting the normal quantum supergroups by twistors which satisfy the graded shifted cocycle condition, thus generalizing the quasi-Hopf twisting procedure to the supersymmetric case. Two types of elliptic quantum supergroups are defined, that is the face type $B_{q,\lambda}(G)$ and the vertex type $A_{q,p}[\hat{sl(n|n)}]$ (and $A_{q,p}[\hat{gl(n|n)}]$), where $G$ is any Kac-Moody superalgebra with symmetrizable generalized Cartan matrix. It appears that the vertex type twistor can be constructed only for $U_q[\hat{sl(n|n)}]$ in a non-standard system of simple roots, all of which are fermionic.Comment: 22 pages, Latex fil

Quantum affine algebras and universal R-matrix with spectral parameter, II

This paper is an extended version of our previous short letter \cite{ZG2} and is attempted to give a detailed account for the results presented in that paper. Let $U_q({\cal G}^{(1)})$ be the quantized nontwisted affine Lie algebra and $U_q({\cal G})$ be the corresponding quantum simple Lie algebra. Using the previous obtained universal $R$-matrix for $U_q(A_1^{(1)})$ and $U_q(A_2^{(1)})$, we determine the explicitly spectral-dependent universal $R$-matrix for $U_q(A_1)$ and $U_q(A_2)$. We apply these spectral-dependent universal $R$-matrix to some concrete representations. We then reproduce the well-known results for the fundamental representations and we are also able to derive for the first time the extreamly explicit and compact formula of the spectral-dependent $R$-matrix for the adjoint representation of $U_q(A_2)$, the simplest nontrival case when the tensor product of the representations is {\em not} multiplicity-free.Comment: 22 page

Guided Open Vocabulary Image Captioning with Constrained Beam Search

Existing image captioning models do not generalize well to out-of-domain images containing novel scenes or objects. This limitation severely hinders the use of these models in real world applications dealing with images in the wild. We address this problem using a flexible approach that enables existing deep captioning architectures to take advantage of image taggers at test time, without re-training. Our method uses constrained beam search to force the inclusion of selected tag words in the output, and fixed, pretrained word embeddings to facilitate vocabulary expansion to previously unseen tag words. Using this approach we achieve state of the art results for out-of-domain captioning on MSCOCO (and improved results for in-domain captioning). Perhaps surprisingly, our results significantly outperform approaches that incorporate the same tag predictions into the learning algorithm. We also show that we can significantly improve the quality of generated ImageNet captions by leveraging ground-truth labels.Comment: EMNLP 201

God and the Problem of Order: The Role of Perfect Information and Absolute Power in Economics, Religion and Law

The “problem of order” is resolved idyllically in Arrow-Debreu general-equilibrium models through the assumption of perfect information, where each actor is god-like, omniscient, and where this omniscience results in the veracity of the first theorem of welfare economics, where an equilibrium is Pareto efficient, where no actor, given her original endowments of alienable and inalienable capital, can improve her position. Analogously, in many religions, God is understood to be omniscient; order emerges through God’s ability to sanction malfeasance costlessly. Ideally, a comparable order would likewise be the consequence of the assumption in both economics and religion of an omnipotent principal. In both economics and religion, equilibria serve as idealized, transcendent, critical standards. In the real, immanent, world, where information is imperfect and no power is absolute, the institutionalization of law is necessary to maintain order. Religion has bequeathed to the legal order moral principles that may legitimate the law and make it binding, while economics, as a theory of incentives in the immanent world, models the imposition of legal sanctions, which penalize actors who approach and violate the law strategically. The law is an institutional order, where both constitutive norms and legal rules are legitimated through “religious” values and supported by “economic,” situational sanctions

A Unified and Complete Construction of All Finite Dimensional Irreducible Representations of gl(2|2)

Representations of the non-semisimple superalgebra $gl(2|2)$ in the standard basis are investigated by means of the vector coherent state method and boson-fermion realization. All finite-dimensional irreducible typical and atypical representations and lowest weight (indecomposable) Kac modules of $gl(2|2)$ are constructed explicitly through the explicit construction of all $gl(2)\oplus gl(2)$ particle states (multiplets) in terms of boson and fermion creation operators in the super-Fock space. This gives a unified and complete treatment of finite-dimensional representations of $gl(2|2)$ in explicit form, essential for the construction of primary fields of the corresponding current superalgebra at arbitrary level.Comment: LaTex file, 23 pages, two references and a comment added, to appear in J. Math. Phy