Differential calculus onISO q(N), quantum Poincaré algebra and q-gravity

We present a general method to deform the inhomogeneous algebras of theB _n,C_n,D_n type, and find the corresponding bicovariant differential calculus. The method is based on a projection fromB _n+1,C_n+1,D_n+1. For example we obtain the (bicovariant) inhomogeneousq-algebraISO _q(N) as a consistent projection of the (bicovariant)q-algebraSO _q(N=2). This projection works for particular multiparametric deformations ofSO(N+2), the so-called minimal deformations. The case ofISO _q(4) is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameterq. The quantum Poincaré Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains theclassical Lorentz algebra. Only the commutation relations involving the momenta depend onq. Finally, we discuss aq-deformation of gravity based on the gauging of thisq-Poincaré algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.     

The hopf algebra of vector fields on complex quantum groups

We derive the equivalence of the complex quantum enveloping algebra and the algebra of complex quantum vector fields for the Lie algebra types A _n , B _n , C _n , and D _n by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals.Humboldt Fellow.     

Inhomogenous Model of Crossing Loops and Multidegrees of Some Algebraic Varieties

We consider a quantum integrable inhomogeneous model based on the Brauer algebra B(1) and discuss the properties of its ground state eigenvector. In particular we derive various sum rules, and show how some of its entries are related to multidegrees of algebraic varieties.     

Galilean conformal and superconformal symmetries

Firstly we discuss briefly three different algebras named as nonrelativistic (NR) conformal: Schr?dinger, Galilean conformal, and infinite algebra of local NR conformal isometries. Further we shall consider in some detail Galilean conformal algebra (GCA) obtained in the limit c???? from relativistic conformal algebraO(d+1, 2) (d-number of space dimensions). Two different contraction limits providing GCA and some recently considered realizations will be briefly discussed. Finally by considering NR contraction of D = 4 superconformal algebra the Galilei conformal superalgebra (GCSA) is obtained, in the formulation using complexWeyl supercharges.     

Reduction of Quantum Systems with Arbitrary First Class Constraints and Hecke Algebras

We propose a method for reduction of quantum systems with arbitrary first-class constraints. An appropriate mathematical setting for the problem is the homology of associative algebras. For every such algebra A and subalgebra B with augmentation ɛ there exists a cohomological complex which is a generalization of the BRST one. Its cohomology is an associative graded algebra Hk ^*(A,B) which we call the Hecke algebra of the triple (A,B,ɛ). It acts in the cohomology space H ^*(B,V) for every left A module V. In particular the zeroth graded component $Hk^{0}(A,B)$ acts in the space of B invariants of $V$ and provides the reduction of the quantum system. Received: 15 June 1998 / Accepted: 25 January 1999     

Truncated harmonic oscillator and parasupersymmetric quantum mechanics

We discuss in detail the parasupersymmetric quantum mechanics of arbitrary order where the parasupersymmetry is between the normal bosons and those corresponding to the truncated harmonic oscillator. We show that even though the parasusy algebra is different from that of the usual parasusy quantum mechanics, still the consequences of the two are identical. We further show that the parasupersymmetric quantum mechanics of arbitrary orderp can also be rewritten in terms ofp supercharges (i.e. all of which obeyQ _i ² =0). However, the Hamiltonian cannot be expressed in a simple form in terms of thep supercharges except in a special case. A model of conformal parasupersymmetry is also discussed and it is shown that in this case, thep supercharges, thep conformal supercharges along with HamiltonianH, conformal generatorK and dilatation generatorD form a closed algebra.     

Pairs of Compatible Associative Algebras,Classical Yang-Baxter Equation and Quiver Representations

Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, a linear deformation of the matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such deformations and construct numerous examples. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures. We also describe an important class of M-structures related to the affine Dynkin diagrams of A, D, E-type. These M-structures and their representations are described in terms of quiver representations.     

Noncoassociative twisting: Its application to κ-Poincaré algebra

It is shown that the use of noncoassociative coproduct allows us to simplify the structure of-Poincaré algebra — it contains nondeformedD=3 Euclidean quasi-bialgebra. We obtain by the dual construction the commuting, nonassociativeD=4 space-time.Presented at the 4th international Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June, 1995I would like to thank Profs. J. Lukierski and V.N. Tolstoy for valuable comments.     

Construction of the elliptic Gaudin system based on Lie algebra

Gaudin model is a very important integrable model in both quantum field theory and condensed matter physics. The integrability of Gaudin models is related to classical r-matrices of simple Lie algebras and semi-simple Lie algebra. Since most of the constructions of Gaudin models works concerned mainly on rational and trigonometric Gaudin algebras or just in a particular Lie algebra as an alternative to the matrix entry calculations often presented, in this paper we give our calculations in terms of a basis of the typical Lie algebra, A _n , B _n , C _n , D _n , and we calculate a classical r-matrix for the elliptic Gaudin system with spin.      

Possible contractions of quantum orthogonal groups

Possible contractions of quantum orthogonal groups which correspond to different choices of primitive elements of Hopf algebra are considered and all allowed contractions in Cayley-Klein scheme are obtained. Quantum deformations of kinematical groups have been investigated and have shown that quantum analogs of (complex) Galilei group G(1, 3) do not exist in our scheme.     

Associated quantum vector bundles and symplectic structure on a quantum space

We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a Hopf algebra H are particular instances of these extensions, and in these cases we are able to define a differential calculus over their associated vector bundles without requiring the use of a (bicovariant) differential structure over H. Moreover, if H is coquasitriangular, it coacts naturally on the associated bundle, and the differential structure is covariant.We apply this construction to the case of the finite quotient of the SL _q(2) function Hopf algebra at a root of unity (q ³ = 1) as the structure group, and a reduced 2-dimensional quantum plane as both the base manifold and fibre, getting an algebra which generalizes the notion of classical phase space for this quantum space. We also build explicitly a differential complex for this phase space algebra, and find that levels 0 and 2 support a (co)representation of the quantum symplectic group. On this phase space we define vector fields, and with the help of the Sp _q structure we introduce a symplectic form relating 1-forms to vector fields. This leads naturally to the introduction of Poisson brackets, a necessary step to do classical mechanics on a quantum space, the quantum plane.     

Differential geometry of the Lie algebra of the quantum plane

We present a differential calculus on the extension of the quantum plane obtained by considering that the (bosonic) generator x is invertible and by working with polynomials in ln x instead of polynomials in x. We construct the quantum Lie algebra associated with this extension and obtain its Hopf algebra structure and its dual Hopf algebra.     

Ground state pressure and energy density of an interacting homogeneous Bose gas in two dimensions

We consider an interacting homogeneous Bose gas at zero temperature in two spatial dimensions. The properties of the system can be calculated as an expansion in powers of g, where g is the coupling constant. We calculate the ground state pressure and the ground state energy density to second order in the quantum loop expansion. The renormalization group is used to sum up leading and subleading logarithms from all orders in perturbation theory. In the dilute limit, the renormalization group improved pressure and energy density are expansions in powers of the T ^2B and T ^2Bln(T ^2B), respectively, where T ^2B is the two-body T-matrix. Received 19 April 2002 Published online 13 August 2002     

On twist quantizations of <Emphasis Type="Italic">D</Emphasis> = 4 Lorentz and Poincare algebras

We use the decomposition of o(3, 1) = sl(2; ℂ)₁ ⊕sl(2; ℂ)₂ in order to describe nonstandard quantum deformation of o(3, 1) linked with Jordanian deformation of sl(2; ℂ). Using the twist quantization technique, we obtain the deformed coproducts and antipodes, which can be expressed in terms of real physical Lorentz generators. We describe the extension of the considered deformation of D = 4 Lorentz algebra to the twist deformation of D = 4 Poincare algebra with dimensionless deformation parameter. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Vector fields on complex quantum groups

Using previous results we construct theq-analogues of the left invariant vector fields of the quantum enveloping algebra corresponding to the complex Lie algebras of typeA _n–1 ,B _n ,C _n , andD _n . These quantum vector fields are functionals over the complex quantum groupA. In the special caseA ₁ it is shown that this Hopf algebra coincides withU _q sl(2, ).     

Universal Schwinger cocycles of current algebras in (D+1)-dimensions: Geometry and physics

We discuss the universal version of the Schwinger terms of current algebra (we call it the universal Schwinger cocycle) forp=3 (herep denotes the class of the Schatten idealI _p , which is related to the (D+1) space-time dimensions byp=(D+1)/2) in detail, and give a conjecture of the general form of the cocycle for anyp. We also discuss the infinite charge renormalizations, the highest weight vector and state vectors forp=3. Last, we give brief comments on the problems caused by the difficulties to construct the measure of infinite-dimensional Grassmann manifolds.     

Loop and Path Spaces and¶Four-Dimensional BF Theories:¶Connections, Holonomies and Observables

We study the differential geometry of principal G-bundles whose base space is the space of free paths (loops) on a manifold M. In particular we consider connections defined in terms of pairs (A,B), where A is a connection for a fixed principal bundle P(M,G) and B is a 2-form on M. The relevant curvatures, parallel transports and holonomies are computed and their expressions in local coordinates are exhibited. When the 2-form B is given by the curvature of A, then the so-called non-abelian Stokes formula follows. For a generic 2-form B, we distinguish the cases when the parallel transport depends on the whole path of paths and when it depends only on the spanned surface. In particular we discuss generalizations of the non-abelian Stokes formula. We study also the invariance properties of the (trace of the) holonomy under suitable transformation groups acting on the pairs (A,B). In this way we are able to define observables for both topological and non-topological quantum field theories of the BF type. In the non-topological case, the surface terms may be relevant for the understanding of the quark-confinement problem. In the topological case the (perturbative) four-dimensional quantum BF-theory is expected to yield invariants of imbedded (or immersed) surfaces in a 4-manifold M. Received: 28 March 1998 / Accepted: 12 September 1998     

Contractions, Deformations and Curvature

The role of curvature in relation with Lie algebra contractions of the pseudo-orthogonal algebras so(p,q) is fully described by considering some associated symmetrical homogeneous spaces of constant curvature within a Cayley–Klein framework. We show that a given Lie algebra contraction can be interpreted geometrically as the zero-curvature limit of some underlying homogeneous space with constant curvature. In particular, we study in detail the contraction process for the three classical Riemannian spaces (spherical, Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three relativistic ((anti-)de Sitter and Minkowskian) spacetimes. Next, from a different perspective, we make use of quantum deformations of Lie algebras in order to construct a family of spaces of non-constant curvature that can be interpreted as deformations of the above nine spaces. In this framework, the quantum deformation parameter is identified as the parameter that controls the curvature of such “quantum” spaces.     

ZN-Graded Noncommutative Differential Calculus

We consider the algebra M _N (C) ofN × N matrices as a cyclic quantum plane.We also analyze the coaction of the quantum group F and the action of its dualquantum algebra H on it. Then we study the decomposition ofM _N (C) in termsof the quantum algebra representations. Finally, we develop the differential algebraof the cyclic group Z _N with d ^N = 0, where Z _N is viewed as the the subalgebraof diagonal N × N complex matrices, and treat the particularcase N = 3.