$gl_q(n)$-Covariant Multimode Oscillators and q-Symmetric States
In this paper the coherent states and q-symmetric states for $gl_q(n)$-covariant multimode oscillator system are investigated.
q-alg↗
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Tetrahedron Reflection Equation
Reflection equation for the scattering of lines moving in half-plane is obtained. The corresponding geometric picture is related with configurations of half-planes touching the boundary plane in 2+1 dimensions. This equation can be obtained as an additional to the tetrahedron equation consistency condition for a modified Zamolodchikov algebra.
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Fusion rules for Quantum Transfer Matrices as a Dynamical System on Grassmann Manifolds
We show that the set of transfer matrices of an arbitrary fusion type for an integrable quantum model obey these bilinear functional relations, which are identified with an integrable dynamical system on a Grassmann manifold (higher Hirota equation). The bilinear relations were previously known for a particular class of transfer matrices corresponding to rectangular Young diagrams. We extend this result for general Young diagrams. A general solution of the bilinear equations is presented.
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Form-factors of exponential fields in the sine-Gordon model
An integral representation for form-factors of exponential fields in the sine-Gordon model is proposed.
hep-th↗
Drinfeld twist for quantum $su(2)$ in the adjoint representation
We give a detailed description of the adjoint representation of Drinfeld's twist element, as well as of its coproduct, for $su_{q}(2)$. We also discuss, as applications, the computation of the universal R-matrix in this representation and the problem of symmetrization of identical-particle states with quantum $su(2)$ symmetry.
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q-Gauge Theory
We examine some issues that arise in the q-deformation of a gauge theory. If the deformation is carried out by replacing the equal time commutators of free fields by the corresponding q-commutators, the resulting propagators are not very much different from those of the undeformed theory as long as one is dealing with weak fields; but the theory still violates causality. If one postulates a q-deformed S matrix, the corresponding q-causal commutator has 2 poles of different strength and the result again amounts to a deformation of the Lorentz group.
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Unified view of multimode algebras with Fock-like representations
A unified view of general multimode oscillator algebras with Fock-like representations is presented.It extends a previous analysis of the single-mode oscillator algebras.The expansion of the $a_ia_j^{\dagger}$ operators is extended to include all normally ordered terms in creation and annihilation operators and we analyze their action on Fock-like states.We restrict ourselves to the algebras compatible with number operators. The connection between these algebras and generalized statistics is analyzed.We demonstrate our approach by considering the algebras obtainable from the generalized Jordan-Wigner transformation, the para-Bose and para-Fermi algebras, the Govorkov "paraquantization" algebra and generalized quon algebra.
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Twisted quantum affine algebras and solutions to the Yang-Baxter equation
We construct spectral parameter dependent R-matrices for the quantized enveloping algebras of twisted affine Lie algebras. These give new solutions to the spectral parameter dependent quantum Yang-Baxter equation.
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A Quantum Gauge Group Approach to the 2D SU(n) WZNW Model
The canonical quantization of the WZNW model provides a complete set of exchange relations in the enlarged chiral state spaces that include the Gauss components of the monodromy matrices. Regarded as new dynamical variables, the elements of the latter cannot be identified -- they satisfy different exchange relations. Accordingly, the two dimensional theory expressed in terms of the left and right movers' fields does not automatically respect monodromy invariance. Continuing our recent analysis of the problem by gauge theory methods we conclude that physical states (on which the two dimensional field acts as a single valued operator) are invariant under the (permuted) coproduct of the left and right $U_q(sl(n))$. They satisfy additional constraints fully described for n=2.
hep-th↗
Quantum Orthogonal Cayley-Klein Groups in Cartesian Basis
The similarity transformations of quantum orthogonal groups are developed and FRT theory is reformulated to the Cartesian basis. The quantum orthogonal Cayley-Klein groups are introduced as the algebra functions over an associative algebra with the nilpotent generators. The quantum orthogonal Cayley-Klein algebras are obtained as the dual objects to the corresponding quantum groups.
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Two-parameter deformation of the Poincaré algebra
We examine a two-parameter ($\hbar ,$ $λ$) deformation of the Poincarè algebra which is covariant under the action of $SL_q(2,C).$ When $λ\rightarrow 0$ it yields the Poincarè algebra, while in the $\hbar\rightarrow 0$ limit we recover the classical quadratic algebra discussed previously in \cite{ssy95}, \cite{sy95}. The analogues of the Pauli-Lubanski vector $w$ and Casimirs $p^2$ and $w^2$ are found and a set of mutually commuting operators is constructed.
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Riemannian Geometry on Quantum Spaces
An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended to complex manifolds. Examples include the quantum sphere, the complex quantum projective spaces and the two-sheeted space.
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On ${\cal U}_h(sl(2))$, ${\cal U}_h(e(3))$ and their Representations
By solving a set of recursion relations for the matrix elements of the ${\cal U}_h(sl(2))$ generators, the finite dimensional highest weight representations of the algebra were obtained as factor representations. Taking a nonlinear combination of the generators of the two copies of the ${\cal U}_h(sl(2))$ algebra, we obtained ${\cal U}_h(so(4))$ algebra. The latter, on contraction, yields ${\cal U}_h(e(3))$ algebra. A nonlinear map of ${\cal U}_h(e(3))$ algebra on its classical analogue $e(3)$ was obtained. The inverse mapping was found to be singular. It signifies a physically interesting situation, where in the momentum basis, a restricted domain of the eigenvalues of the classical operators is mapped on the whole real domain of the eigenvalues of the deformed operators.
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Geometrical foundations of fractional supersymmetry
A deformed $q$-calculus is developed on the basis of an algebraic structure involving graded brackets. A number operator and left and right shift operators are constructed for this algebra, and the whole structure is related to the algebra of a $q$-deformed boson. The limit of this algebra when $q$ is a $n$-th root of unity is also studied in detail. By means of a chain rule expansion, the left and right derivatives are identified with the charge $Q$ and covariant derivative $D$ encountered in ordinary/fractional supersymmetry and this leads to new results for these operators. A generalized Berezin integral and fractional superspace measure arise as a natural part of our formalism. When $q$ is a root of unity the algebra is found to have a non-trivial Hopf structure, extending that associated with the anyonic line. One-dimensional ordinary/fractional superspace is identified with the braided line when $q$ is a root of unity, so that one-dimensional ordinary/fractional supersymmetry can be viewed as invariance under translation along this line. In our construction of fractional supersymmetry the $q$-deformed bosons play a role exactly analogous to that of the fermions in the familiar supersymmetric case.
hep-th↗
Fusion of twisted representations
The comultiplication formula for fusion products of untwisted representations of the chiral algebra is generalised to include arbitrary twisted representations. We show that the formulae define a tensor product with suitable properties, and determine the analogue of Zhu's algebra for arbitrary twisted representations. As an example we study the fusion of representations of the Ramond sector of the N=1 and N=2 superconformal algebra. In the latter case, certain subtleties arise which we describe in detail.
hep-th↗
Poisson Algebra of Differential Forms
We give a natural definition of a Poisson Differential Algebra. Consistence conditions are formulated in geometrical terms. It is found that one can often locally put the Poisson structure on differential calculus in a simple canonical form by a coordinate transformation. This is in analogy with the standard Darboux's theorem for symplectic geometry. For certain cases there exists a realization of the exterior derivative through a certain canonical one-form. All the above are carried out similarly for the case of a complex Poisson Differential Algebra. The case of one complex dimension is treated in details and interesting features are noted. A conclusion is made in the last section.
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hbar-(Yangian) Deformation of Miura Map and Virasoro Algebra
An hbar-deformed Virasoro Poisson algebra is obtained using the Wakimoto realization of the Sugawara operator for the Yangian double DY_\hbar(sl_2)_c at the critical level c=-2.
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q-Uncertainty Relations
As one would anticipate from the realization of the q-commutators by difference operators, the states of maximum localization are smeared delta functions depending on q.
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