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The model of generalized quons is described in a purely algebraic way. Commutation relations and corresponding consistency conditions for our generalized quons system are studied in terms of quantum Weyl algebras. Fock space representation and corresponding scalar product is also given.
Let $(V,Z)$ be a Topological Quantum Field Theory over a field $f$ defined on a cobordism category whose morphisms are oriented $n+1$-manifolds perhaps with extra structure. Let $(M,χ)$ be a closed oriented $n+1$-manifold $M$ with this extra structure together with $χ\in H^1(M).$ Let $M_{\infty}$ denote the infinite cyclic cover of $M$ given by $χ.$ Consider a fundamental domain $E$ for the action of the integers on $M_{\infty}$ bounded by lifts of a surface $Σ$ dual to $χ,$ and in general position. $E$ can be viewed as a cobordism from $Σ$ to itself. We give Turaev and Viro's proof of their theorem that the similarity class of the non-nilpotent part of $Z(E)$ is an invariant. We give a method to calculate this invariant for the $(V_p,Z_p)$ theories of Blanchet,Habegger, Masbaum and Vogel when $M$ is zero framed surgery to $S^3$ along a knot K. We give a formula for this invariant when $K$ is a twisted double of another knot. We obtain formulas for the quantum invariants of branched covers of knots, and unbranched covers of 0-surgery to $S^3$ along knots.
A non-linear map is applied onto the (non-standard) null-plane deformation of (3+1) Poincaré algebra giving rise to a simpler form of this triangular quantization. A universal $R$-matrix for the null plane quantum algebra is then obtained from a universal $T$-matrix corresponding to a Hopf subalgebra. Finally, the associated Poincaré Poisson--Lie group is quantized by using the FRT approach.
We propose an explicit expression for vacuum expectation values of the boundary field e^{iaϕ_{B}} in the boundary sine-Gordon model with zero bulk mass. This expression agrees with known exact results for the boundary free energy and with perturbative calculations.
We construct a Chern-Simons action for q-deformed gauge theory which is a simple and straightforward generalization of the usual one. Space-time continues to be an ordinary (commuting) manifold, while the gauge potentials and the field strengths become q-commuting fields. Our approach, which is explicitly carried out for the case of `minimal' deformations, has the advantage of leading naturally to a consistent Hamiltonian structure that has essentially all of the features of the undeformed case. For example, using the new Poisson brackets, the constraints form a closed algebra and generate q-deformed gauge transformations.
The semiclassical approximation for the partition function in Chern-Simons gauge theory is derived using the invariant integration method. Volume and scale factors which were undetermined and had to be fixed by hand in previous derivations are automatically taken account of in this framework. Agreement with Witten's exact expressions for the partition function in the weak coupling (large k) limit is verified for gauge group SU(2) and spacetimes S^3, S^2 x S^1, S^1 x S^1 x S^1 and L(p,q).
We apply the ideas of Alvarez and Labastida to the invariant of spin networks defined by Witten and Martin based on Chern-Simons theory. We show that it is possible to define ambient invariants of spin networks that (for the case of SU(2)) can be considered as extensions to spin networks of the Jones polynomial. Expansions of the coefficients of the polynomial yield primitive Vassiliev invariants. The resulting invariants are candidates for solutions of the Wheeler--DeWitt equations in the spin network representation of quantum gravity.
We discuss the problem of factorisation of the symmetric Macdonald polynomials and present the obtained results for the cases of 2 and 3 variables.
In this paper we define an algebra structure on the vector space $L(Σ)$ generated by links in the manifold $Σ\times [0,1]$ where $Σ$ is an oriented surface. This algebra has a filtration and the associated graded algebra $L_{Gr}(Σ)$ is naturally a Poisson algebra. There is a Poisson algebra homomorphism from the algebra of chord diagrams $ch(Σ)$ on $Σ$ to $L_{Gr}(Σ)$. We show that multiplication in $L(Σ)$ provides a geometric way to define a deformation quantization of the algebra of chord diagrams, provided there is a universal Vassiliev invariant for links in $Σ\times [0,1]$. The quantization descends to a quantization of the moduli space of flat connections on $Σ$ and it is universal with respect to group homomorphisms. If $Σ$ is compact with free fundamental group we construct a universal Vassiliev invariant.
Let $Γ$ be an $N^2$-dimensional bicovariant first order differential calculus on a Hopf algebra $SL_q(N)$. There are three possibilities to construct a differential Z-graded Hopf algebra $Γ^\wedge$ which contains $Γ$ as its first order part. Let $q$ be a transcendental complex number. For $N>2$ these three Z-graded Hopf algebras coincide. For Woronowicz' external algebra we calculate the dimensions of the spaces of left-invariant and bi-invariant $k$-forms. In this case each bi-invariant form is closed. In case of $4D_\pm$ calculi on $SL_q(2)$ the universal calculus is strictly larger than the other two calculi. In particular, the bi-invariant 1-form is not closed.
This is a continuation of our "Lecture on Kac--Moody Lie algebras of the arithmetic type" \cite{25}. We consider hyperbolic (i.e. signature $(n,1)$) integral symmetric bilinear form $S:M\times M \to {\Bbb Z}$ (i.e. hyperbolic lattice), reflection group $W\subset W(S)$, fundamental polyhedron $\Cal M$ of $W$ and an acceptable (corresponding to twisting coefficients) set $P({\Cal M})\subset M$ of vectors orthogonal to faces of $\Cal M$ (simple roots). One can construct the corresponding Lorentzian Kac--Moody Lie algebra ${\goth g}={\goth g}^{\prime\prime}(A(S,W,P({\Cal M})))$ which is graded by $M$. We show that $\goth g$ has good behavior of imaginary roots, its denominator formula is defined in a natural domain and has good automorphic properties if and only if $\goth g$ has so called {\it restricted arithmetic type}. We show that every finitely generated (i.e. $P({\Cal M})$ is finite) algebra ${\goth g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1)))$ may be embedded to ${\goth g}^{\prime\prime}(A(S,W,P({\Cal M})))$ of the restricted arithmetic type. Thus, Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a natural class to study. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the best automorphic properties for the denominator function if they have {\it a lattice Weyl vector $ρ$}. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type with generalized lattice Weyl vector $ρ$ are called {\it elliptic}
We find automorphic corrections for the Lorentzian Kac--Moody algebras with the simplest generalized Cartan matrices of rank 3: A_{1,0} = 2 0 -1 0 2 -2 -1 -2 2 and A_{1,I} = 2 -2 -1 -2 2 -1 -1 -1 2 For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form χ_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30 with respect to a 2-congruence subgroup. We find infinite product or sum expansions for these forms. Our method of construction of χ_{35} leads to the direct construction of Siegel modular forms by infinite product expansions, whose divisors are the Humbert surfaces with fixed discriminants. Existence of these forms was proved by van der Geer in 1982 using some geometrical consideration. We announce a list of all hyperbolic symmetric generalized Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a lattice Weyl vector, and A contains the affine submatrix \tilde{A}_1.
The correct Hamiltonian for an extended Hubbard model with quantum group symmetry as introduced by A. Montorsi and M. Rasetti is derived for a D-dimensional lattice. It is shown that the superconducting SUq(2) holds as a true quantum symmetry only for D = 1 and that terms of higher order in the fermionic operators in addition to phonons are required for a quantum symmetric hamiltonian. The condition for quantum symmetry is "half filling" and there is no local electron-phonon coupling. A discussion of Quantum symmetries in general is given in a formalism that should be readily accessible to non Hopf-algebraists.
We study models of quantum statistical mechanics which can be solved by the algebraic Bethe ansatz. The general method of calculation of correlation functions is based on the method of determinant representations. The auxiliary Fock space and auxiliary Bose fields are introduced in order to remove the two body scattering and represent correlation functions as a mean value of a determinant of a Fredholm integral operator; the representation has a simple form for large space and time separations. In this paper we explain how to calculate the mean value in the auxiliary Fock space of asymptotic expression of the Fredholm determinant. It is necessary for the evaluation of the asymptotic form of the physical correlation functions.
We explicitly evaluate the infinite series of integrals that appears in the "Anderson-Yuval" reformulation of the anisotropic Kondo problem in terms of a one-dimensional Coulomb gas. We do this by developing a general approach relating the anisotropic Kondo problem of arbitrary spin with the boundary sine-Gordon model, which describes impurity tunneling in a Luttinger liquid and in the fractional quantum Hall effect. The Kondo solution then follows from the exact perturbative solution of the latter model in terms of Jack polynomials.
We apply a recently proposed definition of a linear connection in non commutative geometry based on the natural bimodule structure of the algebra of differential forms to the case of the two-parameter quantum plane. We find that there exists a non trivial family of linear connections only when the two parameters obeys a specific relation.
It is shown that q-deformed quantum mechanics (q-deformed Heisenberg algebra) can be interpreted as quantum mechanics on Kaehler manifolds, or as a quantum theory with second (or first-) class constraints. (Saclay, T93/027).
We discuss an interrelation between quantum integrable models and classical soliton equations with discretized time. It appeared that spectral characteristics of quantum integrable systems may be obtained from entirely classical set up. Namely, the eigenvalues of the quantum transfer matrix and the scattering $S$-matrix itself are identified with a certain $τ$-functions of the discrete Liouville equation. The Bethe ansatz equations are obtained as dynamics of zeros. For comparison we also present the Bethe ansatz equations for elliptic solutions of the classical discrete Sine-Gordon equation. The paper is based on the recent study of classical integrable structures in quantum integrable systems, hep-th/9604080.