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Some aspects of differential and integral calculi on generalized grassmann (paragrassmann) algebras are considered. The integration over paragrassmann variables is applied to evaluate the partition function for the $Z_{p+1}$ Potts model on a chain. Finite dimensional paragrassmann representations for $GL_{q}(2)$ are constructed.
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.
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.
Inspired by Reshetikhin's twisting procedure to obtain multiparametric extensions of a Hopf algebra, a general `symmetry transformation' of the `particle conserving' $R$-matrix is found such that the resulting multiparametric $R$-matrix, with a spectral parameter as well as a colour parameter, is also a solution of the Yang-Baxter equation (YBE). The corresponding transformation of the quantum YBE reveals a new relation between the associated quantized algebra and its multiparametric deformation. As applications of this general relation to some particular cases, multiparametric and coloured extensions of the quantum group $GL_q(N)$ and the Yangian algebra $Y(gl_N)$ are investigated and their explicit realizations are also discussed. Possible interesting physical applications of such extended Yangian algebras are indicated.
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.
The realizations of the Lie algebra corresponding to the dynamical symmetry group SO(2,1) of the Schrödinger equations for the Morse and the $V=u^2+1/u^2$ potentials were known to be related by a canonical transformation. q-deformed analog of this transformation connecting two different realizations of the sl_q(2) algebra is presented. By the virtue of the q-canonical transformation a q-deformed Schrödinger equation for the Morse potential is obtained from the q-deformed $V=u^2+ 1/u^2$ Schrödinger equation. Wave functions and eigenvalues of the q-Schrödinger equations yielding a new definition of the q-Laguerre polynomials are studied.
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.
We supplement the discussion of Moore and Reshetikhin and others by finding new semiclassical nonabelian vertex operators for the chiral, antichiral and nonchiral primary fields of WZW theory. These new nonabelian vertex operators are the natural generalization of the familiar abelian vertex operators: They involve only the representation matrices of Lie $g$, the currents of affine $(g \times g)$ and certain chiral and antichiral zero modes, and they reduce to the abelian vertex operators in the limit of abelian algebras. Using the new constructions, we also discuss semiclassical operator product expansions, braid relations and relations to the known form of the semiclassical affine-Sugawara conformal blocks.
New identities on traces of representations of the Hecke algebra on the spaces of paths on graphs are presented. These identities are relevant in the computation of partition functions with fixed boundary conditions and of two-point correlation functions in a class of lattice models. They might also help to determine the Boltzmann weights of these (allegedly) integrable models.
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.
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.
For the vacuum sectors of regime-III ABF models, we observe that two sets of combinatorial objects: the strings which parametrize the row-to-row transfer matrix eigenvectors, and the paths which parametrize the corner transfer matrix eigenvectors, can both be expressed in terms of the same set of standard tableaux. Furthermore, the momenta of the strings, the energies of the paths, and the charges of the tableaux are such that there is a weight-preserving bijection between the two sets of eigenvectors, wherein the tableaux play an interpolating role. This bijection is so natural, that we conjecture that it exists in general.
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.
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.
We find that there is an alternative possibility to define the chirality operator on the fuzzy sphere, due to the ambiguity of the operator ordering. Adopting this new chirality operator and the corresponding Dirac operator, we define Connes' spectral triple on the fuzzy sphere and the differential calculus. The differential calculus based on this new spectral triple is simplified considerably. Using this formulation the action of the scalar field is derived.
The usual description of 2+1 dimensional Einstein gravity as a Chern-Simons (CS) theory is extended to a one parameter family of descriptions of 2+1 Einstein gravity. This is done by replacing the Poincare' gauge group symmetry by a q-deformed Poincare' gauge group symmetry, with the former recovered when q-> 1. As a result, we obtain a one parameter family of Hamiltonian formulations for 2+1 gravity. Although formulated in terms of noncommuting dreibeins and spin-connection fields, our expression for the action and our field equations, appropriately ordered, are identical in form to the ordinary ones. Moreover, starting with a properly defined metric tensor, the usual metric theory can be built; the Christoffel symbols and space-time curvature having the usual expressions in terms of the metric tensor, and being represented by c-numbers. In this article, we also couple the theory to particle sources, and find that these sources carry exotic angular momentum. Finally, problems related to the introduction of a cosmological constant are discussed.
The 27-dimensional Hopf algebra A(F), defined by the exact sequence of quantum groups A(SL(2,C))->A(SL_q(2))->A(F), q^3=1, is studied as a finite quantum group symmetry of the matrix algebra M(3,C), describing the color sector of Alain Connes' formulation of the Standard Model. The duality with the Hopf algebra H,investigated in a recent work by Robert Coquereaux, is established and used to define a representation of H on M(3,C) and two commuting representations of H on A(F).
Dynamics has been generalized to a noncommutative phase space. The noncommuting phase space is taken to be invariant under the quantum group $GL_{q,p}(2)$. The $q$-deformed differential calculus on the phase space is formulated and using this, both the Hamiltonian and Lagrangian forms of dynamics have been constructed. In contrast to earlier forms of $q$-dynamics, our formalism has the advantage of preserving the conventional symmetries such as rotational or Lorentz invariance.