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

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

Identical Particles and Quantum Symmetries

We propose a solution to the problem of compatibility of Bose-Fermi statistics with symmetry transformations implemented by compact quantum groups of Drinfel'd type. We use unitary transformations to conjugate multi-particle symmetry postulates, so as to obtain a twisted realization of the symmetric groups S_n.Comment: latex, 30 pages; final version (Nucl. Phys. B, in print

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

Spinon Bases, Yangian Symmetry and Fermionic Representations of Virasoro Characters in Conformal Field Theory

We study the description of the $SU(2)$, level $k=1$, Wess-Zumino-Witten conformal field theory in terms of the modes of the spin-1/2 affine primary field $\phi^\alpha$. These are shown to satisfy generalized `canonical commutation relations', which we use to construct a basis of Hilbert space in terms of representations of the Yangian $Y(sl_2)$. Using this description, we explicitly derive so-called `fermionic representations' of the Virasoro characters, which were first conjectured by Kedem et al.~\cite{kedem}. We point out that similar results are expected for a wide class of rational conformal field theories.Comment: 15 pages, LaTeX, USC-94/4, PUPT-146

Classical Yang-Baxter Equation and Low Dimensional Triangular Lie Bialgebras

All solutions of constant classical Yang-Baxter equation (CYBE) in Lie algebra $L$ with dim $L \le 3$ are obtained and the sufficient and necessary conditions which $(L, \hbox {[ ]}, \Delta_r, r)$ is a coboundary (or triangular) Lie bialgebra are given. The strongly symmetric elements in $L\otimes L$ are found and they all are solutions of CYBE in $L$ with $dim L \le 3$.Comment: 17page

The Quantum Double in Integrable Quantum Field Theory

Various aspects of recent works on affine quantum group symmetry of integrable 2d quantum field theory are reviewed and further clarified. A geometrical meaning is given to the quantum double, and other properties of quantum groups. Multiplicative presentations of the Yangian double are analyzed.Comment: 43 page

Coadjoint Orbits of the Generalised Sl(2) Sl(3) Kdv Hierarchies

In this paper we develop two coadjoint orbit constructions for the phase spaces of the generalised $Sl(2)$ and $Sl(3)$ KdV hierachies. This involves the construction of two group actions in terms of Yang Baxter operators, and an Hamiltonian reduction of the coadjoint orbits. The Poisson brackets are reproduced by the Kirillov construction. From this construction we obtain a `natural' gauge fixing proceedure for the generalised hierarchies.Comment: 37 page

Discretization of Virasoro Algebra

A $q$-discretization of \vi\ algebra is studied which reduces to the ordinary \vi\ algebra in the limit of q \ra 1. This is derived starting from the Moyal bracket algebra, hence is a kind of quantum deformation different from the quantum groups. Representation of this new algebra by using $q$-parametrized free fields is also given.Comment: 12 pages, Latex, TMUP-HEL-930

On Soliton Content of Self Dual Yang-Mills Equations

Exploiting the formulation of the Self Dual Yang-Mills equations as a Riemann-Hilbert factorization problem, we present a theory of pulling back soliton hierarchies to the Self Dual Yang-Mills equations. We show that for each map \C^4 \to \C^{\infty } satisfying a simple system of linear equations formulated below one can pull back the (generalized) Drinfeld-Sokolov hierarchies to the Self Dual Yang-Mills equations. This indicates that there is a class of solutions to the Self Dual Yang-Mills equations which can be constructed using the soliton techniques like the $\tau$ function method. In particular this class contains the solutions obtained via the symmetry reductions of the Self Dual Yang-Mills equations. It also contains genuine 4 dimensional solutions . The method can be used to study the symmetry reductions and as an example of that we get an equation exibiting breaking solitons, formulated by O. Bogoyavlenskii, as one of the $2 + 1$ dimensional reductions of the Self Dual Yang-Mills equations.Comment: 11 pages, plain Te