An Unfolded Quantization for Twisted Hopf Algebras

In this talk I discuss a recently developed "Unfolded Quantization Framework". It allows to introduce a Hamiltonian Second Quantization based on a Hopf algebra endowed with a coproduct satisfying, for the Hamiltonian, the physical requirement of being a primitive element. The scheme can be applied to theories deformed via a Drinfeld twist. I discuss in particular two cases: the abelian twist deformation of a rotationally invariant nonrelativistic Quantum Mechanics (the twist induces a standard noncommutativity) and the Jordanian twist of the harmonic oscillator. In the latter case the twist induces a Snyder non-commutativity for the space-coordinates, with a pseudo-Hermitian deformed Hamiltonian. The "Unfolded Quantization Framework" unambiguously fixes the non-additive effective interactions in the multi-particle sector of the deformed quantum theory. The statistics of the particles is preserved even in the presence of a deformation.Comment: 9 pages. Talk given at QTS7 (7th Int. Conf. on Quantum Theory and Symmetries, Prague, August 2011

Y(so(5)) symmtry of the nonlinear Schro¨\ddot{o}dinger model with four-cmponents

The quantum nonlinear Schr$\ddot{o}$dinger(NLS) model with four-component fermions exhibits a $Y(so(5))$ symmetry when considered on an infintite interval. The constructed generators of Yangian are proved to satisfy the Drinfel'd formula and furthermore, the $RTT$ relation with the general form of rational R-matrix given by Yang-Baxterization associated with $so(5)$ algebraic structure.Comment: 10 pages, no figure

Learning from Julius' star, *, ⋆\star

While collecting some personal memories about Julius Wess, I briefly describe some aspects of my recent work on many particle quantum mechanics and second quantization on noncommutative spaces obtained by twisting, and their connection to him.Comment: Late2e file 13 pages. To appear in the Proceedings of the Workshop "Scientific and Human Legacy of Julius Wess - JW2011", Donji Milanovac (Serbia), August 27-29, 2011, International Journal of Modern Physics: Conference Series. On-line at: http://www.worldscientific.com/toc/ijmpcs/13/0

Twisted Quantum Fields on Moyal and Wick-Voros Planes are Inequivalent

The Moyal and Wick-Voros planes A^{M,V}_{\theta} are *-isomorphic. On each of these planes the Poincar\'e group acts as a Hopf algebra symmetry if its coproducts are deformed by twist factors. We show that the *-isomorphism T: A^M_{\theta} to A^V_{\theta} does not also map the corresponding twists of the Poincar\'e group algebra. The quantum field theories on these planes with twisted Poincar\'e-Hopf symmetries are thus inequivalent. We explicitly verify this result by showing that a non-trivial dependence on the non-commutative parameter is present for the Wick-Voros plane in a self-energy diagram whereas it is known to be absent on the Moyal plane (in the absence of gauge fields). Our results differ from these of (arXiv:0810.2095 [hep-th]) because of differences in the treatments of quantum field theories.Comment: 12 page

Minimal Affinizations of Representations of Quantum Groups: the simply--laced case

We continue our study of minimal affinizations for algebras of type D, E.Comment: 25 page

Integrals of motion of the Haldane Shastry Model

In this letter we develop a method to construct all the integrals of motion of the $SU(p)$ Haldane-Shastry model of spins, equally spaced around a circle, interacting through a $1/r^2$ exchange interaction. These integrals of motion respect the Yangian symmetry algebra of the Hamiltonian.Comment: 13 pages, REVTEX v3.

A central extension of \cD Y_{\hbar}(\gtgl_2) and its vertex representations

A central extension of \cD Y_{\hbar}(\gtgl_2) is proposed. The bosonization of level $1$ module and vertex operators are also given.Comment: 10 pages, AmsLatex, to appear in Lett. in Math. Phy

Super Yangian Double DY(gl(1∣1))DY( gl(1|1)) and Its Gauss Decomposition

We extend Yangian double to super (or graded) case and give its Drinfel'd generators realization by Gauss decomposition.Comment: 6 pages, Latex, no figure

Worldsheet boundary conditions in Poisson-Lie T-duality

We apply canonical Poisson-Lie T-duality transformations to bosonic open string worldsheet boundary conditions, showing that the form of these conditions is invariant at the classical level, and therefore they are compatible with Poisson-Lie T-duality. In particular the conditions for conformal invariance are automatically preserved, rendering also the dual model conformal. The boundary conditions are defined in terms of a gluing matrix which encodes the properties of D-branes, and we derive the duality map for this matrix. We demonstrate explicitly the implications of this map for D-branes in two non-Abelian Drinfel'd doubles.Comment: 20 pages, Latex; v2: typos and wording corrected, references added; v3: three-dimensional example added, reference added, discussion clarified, published versio

The structure of quantum Lie algebras for the classical series B_l, C_l and D_l

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at $q=1$. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for adjoint x adjoint ---> adjoint. We present a practical method for the determination of these quantum Clebsch-Gordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum Lie algebras associated to the classical Lie algebras B_l, C_l and D_l. In the quantum case also the structure constants of the Cartan subalgebra are non-zero and we observe that they are determined in terms of the simple quantum roots. We introduce an invariant Killing form on the quantum Lie algebras and find that it takes values which are simple q-deformations of the classical ones.Comment: 25 pages, amslatex, eepic. Final version for publication in J. Phys. A. Minor misprints in eqs. 5.11 and 5.12 correcte