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At least 433 records · Page 24Linked to original sources
arXiv1997

Moebius Transformations in Noncommutative Conformal Geometry

We study the projective linear group PGL_2(A), associated with an arbitrary algebra A, and its subgroups from the point of view of their action on the space of involutions in A. This action formally resembles Moebius transformations known from complex geometry. By specifying A to be an algebra of bounded operators in a Hilbert space H, we rediscover the Moebius group defined by Connes and study its action on the space of Fredholm modules over the algebra A. There is an induced action on the K-homology of A, which turns out to be trivial. Moreover, this action leads naturally to a simpler object, the polarized module underlying a given Fredholm module, and we discuss this relation in detail. Any polarized module can be lifted to a Fredholm module, and the set of different lifts forms a category, whose morphisms are given by generalized Moebius tranformations. We present an example of a polarized module canonically associated with the differentiable structure of a smooth manifold V. Using our lifting procedure we obtain a class of Fredholm modules characterizing the conformal structures on V. Fredholm modules obtained in this way are a special case of those constructed by Connes, Sullivan and Teleman.

math-ph↗
arXiv1997

A lecture on Arithmetic Mirror Symmetry and Calabi-Yau manifolds

We extend our variant of mirror symmetry for K3 surfaces \cite{GN3} and clarify its relation with mirror symmetry for Calabi-Yau manifolds. We introduce two classes (for the models A and B) of Calabi-Yau manifolds fibrated by K3 surfaces with some special Picard lattices. These two classes are related with automorphic forms on IV type domains which we studied in our papers \cite{GN1}-\cite{GN6}. Conjecturally these automorphic forms take part in the quantum intersection pairing for model A, Yukawa coupling for model B and mirror symmetry between these two classes of Calabi-Yau manifolds. Recently there were several papers by physicists where it was shown on some examples. We propose a problem of classification of introduced Calabi-Yau manifolds. Our papers \cite{GN1}-\cite{GN6} and \cite{N3}-\cite{N14} give a hope that this is possible. They describe possible Picard or transcendental lattices of general K3 fibers of the Calabi-Yau manifolds.

alg-geom↗
arXiv1995

Self-dual Koornwinder-Macdonald polynomials

We prove certain duality properties and present recurrence relations for a four-parameter family of self-dual Koornwinder-Macdonald polynomials. The recurrence relations are used to verify Macdonald's normalization conjectures for these polynomials.

q-alg↗
arXiv1995

Geometry of the Quantum Complex Projective Space $CP_q(N)$

The quantum deformation $CP_q(N)$ of complex projective space is discussed. Many of the features present in the case of the quantum sphere can be extended. The differential and integral calculus is studied and $CP_q(N)$ appears as a quantum Kähler manifold. The braiding of several copies of $CP_q(N)$ is introduced and the anharmonic ratios of four collinear points are shown to be invariant under quantum projective transformations. They provide the building blocks of all projective invariants. The Poisson limit is also described.

q-alg↗
arXiv1997

Twisted h-spacetimes and invariant equations

We analyze the h-deformations of the Lorentz group and their associated spacetimes. We prove that they have a twisted character and give explicitly the twisting matrices. After studying the representations of one of the deformed spacetime algebras, we discuss the Klein-Gordon operator. It is found that the h-deformed d'Alembertian has plane wave solutions of the same form as the standard ones. We also give explicit expressions for the h-gamma matrices defining the associated Dirac equations.

q-alg↗
arXiv1997

Some comments on $q$-deformed oscillators and $q$-deformed $su(2)$ algebras

The various relations between $q$-deformed oscillators algebras and the $q$-deformed $su(2)$ algebras are discussed. In particular, we exhibit the similarity of the $q$-deformed $su(2)$ algebra obtained from $q$-oscillators via Schwinger construction and those obtained from $q$-Holstein-Primakoff transformation and show how the relation between $su_{\sqrt{q}}(2)$ and Hong Yan $q$-oscillator can be regarded as an special case of Inöuë- Wigner contraction. This latter observation and the imposition of positive norm requirement suggest that Hong-Yan $q$-oscillator algebra is different from the usual $su_{\sqrt{q}}(2)$ algebra, contrary to current belief in the literature.

q-alg↗
arXiv1995

Field Theory on $q=-1$ Quantum Plane

We build the $q=-1$ defomation of plane on a product of two copies of algebras of functions on the plane. This algebra constains a subalgebra of functions on the plane. We present general scheme (which could be used as well to construct quaternion from pairs of complex numbers) and we use it to derive differential structures, metric and discuss sample field theoretical models.

q-alg↗
arXiv1997

Quantization of the Algebra of Chord Diagrams

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.

q-alg↗
arXiv1997

Higher Order Differential Calculus on $SL_q(N)$

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.

q-alg↗
arXiv1995

Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras

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}

alg-geom↗
arXiv1996

The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras

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.

alg-geom↗
arXiv1995

On Quantum Groups in the Hubbard Model with Phonons

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.

cond-mat↗
arXiv1997

Normal Ordering in the Theory of Correlation Functions of Exactly Solvable Models

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.

hep-th↗
arXiv1995

Exact perturbative solution of the Kondo problem

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.

cond-mat↗
arXiv1995

Linear Connections on the Two Parameter Quantum Plane

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.

q-alg↗