On the Idempotents of Hecke Algebras

We give a new construction of primitive idempotents of the Hecke algebras associated with the symmetric groups. The idempotents are found as evaluated products of certain rational functions thus providing a new version of the fusion procedure for the Hecke algebras. We show that the normalization factors which occur in the procedure are related to the Ocneanu–Markov trace of the idempotents.      

On the Hochschild Homology of Elliptic Sklyanin Algebras

In this paper, we compute the Hochschild homology of elliptic Sklyanin algebras. These algebras are deformations of polynomial algebra with a Poisson bracket called the Sklyanin Poisson bracket.      

Superfield Formulation for Non-Relativistic Chern–Simons-Matter Theory

We construct a superfield formulation for non-relativistic Chern–Simons-matter theories with manifest dynamical supersymmetry. By eliminating all the auxiliary fields, we show that the simple action reduces to the one obtained by taking non-relativistic limit from the relativistic Chern–Simons-matter theory proposed in the literature. As a further application, we give a manifestly supersymmetric derivation of the non-relativistic ABJM theory.      

Spectral Asymptotic for the High-Dimensional Euler Top

In this paper, we compute the leading coefficient in the asymptotic expansion of the joint eigenvalues for the high-dimensional Euler Top. We also prove a central limit theorem for the same eigenvalues.      

On a Selberg–Schur Integral

A generalization of Selberg’s beta integral involving Schur polynomials associated with partitions with entries not greater than 2 is explicitly computed. The complex version of this integral is given after proving a general statement concerning the complex extensions of Selberg–Schur integrals. All these results have interesting applications in both mathematics and physics, particularly in conformal field theory, because the conformal blocks for the Wess–Zumino–Novikov–Witten model with underlying affine structure can be obtained by analytical continuation of these integrals.      

Polyakov Soldering and Second-Order Frames: The Role of the Cartan Connection

The so-called ‘soldering’ procedure performed by Polyakov (Int J Math Phys A5, 833–842, 1990) for a -gauge theory is geometrically explained in terms of a Cartan connection on second-order frames of the projective space P¹. The relationship between a Cartan connection and the usual (Ehresmann) connection on a principal bundle allows to gain an appropriate insight into the derivation of the genuine ‘diffeomorphisms out of gauge transformations’ given by Polyakov himself. Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix-Marseille I, Aix-Marseille II et de l’Université du Sud Toulon-Var. Unité affiliée à la FRUMAM Fédération de Recherche 2291.     

Consistency of Regularization for Free Scalar Fields

In two dimensional constructive quantum field theory for scalar fields, it is necessary to regularize both the action and the total (Gaussian) volume. In this paper we consider the compatibility of these regularizations.      

On the Uniqueness of Weak Weyl Representations of the Canonical Commutation Relation

Let (T, H) be a weak Weyl representation of the canonical commutation relation (CCR) with one degree of freedom. Namely T is a symmetric operator and H is a self-adjoint operator on a complex Hilbert space satisfying the weak Weyl relation: for all (the set of real numbers), e^−itH D(T) ⊂ D(T) (i is the imaginary unit and D(T) denotes the domain of T) and . In the context of quantum theory where H is a Hamiltonian, T is called a strong time operator of H. In this paper we prove the following theorem on uniqueness of weak Weyl representations: Let be separable. Assume that H is bounded below with and , where is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Then ( is the closure of T) is unitarily equivalent to a direct sum of the weak Weyl representation on the Hilbert space , where is the multiplication operator by the variable and with . Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation . This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from Japan Society for the Promotion of Science (JSPS).     

Semi-Classical Quantum Fields Theories and Frobenius Manifolds

We show that a semi-classical quantum field theory comes with a versal family with the property that the corresponding partition function generates all path integrals., satisfies a system of second order differential equations determined by algebras of classical observables. This versal family gives rise to a notion of special coordinates that is analogous to that in string theories. We also show that for a large class of semi-classical theories, their moduli space has the structure of a Frobenius super-manifold. This work was supported by KOSEF Interdisciplinary Research Grant No. R01-2006-000-10638-0.     

Kramers Degeneracy Theorem in Nonrelativistic QED

Degeneracy of the eigenvalues of the Pauli–Fierz Hamiltonian with spin 1/2 is proven by the Kramers degeneracy theorem. The Pauli–Fierz Hamiltonian at fixed total momentum is also investigated.      

On the Instability Intervals of the Mathieu–Hill Operator

Consider the Mathieu–Hill operator

Rota–Baxter Algebras and New Combinatorial Identities

The word problem for an arbitrary associative Rota–Baxter algebra is solved. This leads to a noncommutative generalization of the classical Spitzer identities. Links to other combinatorial aspects are indicated.      

Complete Set of Eigenfunctions of the Quantum Toda Chain

The integral representation of the eigenfunctions of quantum periodic Toda chain constructed by Kharchev and Lebedev is revisited. We prove that Pasquier and Gaudin’s solutions of the Baxter equation provides a complete set of eigenfunctions under this integral representation. This will, in addition, show that the joint spectrum of commuting Hamiltonians of the quantum periodic Toda chain is simple.      

Quantum Dynamical Applications of Salem’s Theorem

We consider the survival probability of a state that evolves according to the Schrödinger dynamics generated by a self-adjoint operator H. We deduce from a classical result of Salem that upper bounds for the Hausdorff dimension of a set supporting the spectral measure associated with the initial state imply lower bounds on a subsequence of time scales for the survival probability. This general phenomenon is illustrated with applications to the Fibonacci operator and the critical almost Mathieu operator. In particular, this gives the first quantitative dynamical bound for the critical almost Mathieu operator.     

Dynamical Noncommutativity and Noether Theorem in Twisted $${\phi^{\star 4}}$$ Theory

A -product is defined via a set of commuting vector fields , and used in a theory coupled to the fields. The -product is dynamical, and the vacuum solution , reproduces the usual Moyal product. The action is invariant under rigid translations and Lorentz rotations, and the conserved energy–momentum and angular momentum tensors are explicitly derived.      

Classification and Realizations of Type III Factor Representations of Cuntz–Krieger Algebras Associated with Quasi-Free States

We completely classify type III factor representations of Cuntz–Krieger algebras associated with quasi-free states up to unitary equivalence. Furthermore, we realize these representations on concrete Hilbert spaces without using GNS construction. Free groups and their type II₁ factor representations are used in these realizations.      

Fermionic Realization of Two-Parameter Quantum Affine Algebra $${U_{r,s}(\widehat{\mathfrak {sl}_n})}$$

We construct all fundamental modules for the two parameter quantum affine algebra of type A using a combinatorial model of Young diagrams. In particular, we also give a fermionic realization of the two-parameter quantum affine algebra.      

Cohomology of $${\mathfrak{osp}}(1|2)$$ Acting on Linear Differential Operators on the Supercircle S1|1

We compute the first cohomology spaces of the Lie superalgebra with coefficients in the superspace of linear differential operators acting on weighted densities on the supercircle S ^1|1. The structure of these spaces was conjectured in (Gargoubi et al. in Lett Math Phys 79:5165, 2007). In fact, we prove here that the situation is a little bit more complicated.      

Integrable Discrete Nets in Grassmannians

We consider discrete nets in Grassmannians , which generalize Q-nets (maps with planar elementary quadrilaterals) and Darboux nets (-valued maps defined on the edges of such that quadruples of points corresponding to elementary squares are all collinear). We give a geometric proof of integrability (multidimensional consistency) of these novel nets, and show that they are analytically described by the noncommutative discrete Darboux system.