On two theorems for flat, affine group schemes over a discrete valuation ring

We include short and elementary proofs of two theorems characterizing reductive group schemes over a discrete valuation ring, in a slightly more general context.Comment: 10 pages. To appear in C. E. J.

On S-duality for Non-Simply-Laced Gauge Groups

We point out that for N=4 gauge theories with exceptional gauge groups G_2 and F_4 the S-duality transformation acts on the moduli space by a nontrivial involution. We note that the duality groups of these theories are the Hecke groups with elliptic elements of order six and four, respectively. These groups extend certain subgroups of SL(2,Z) by elements with a non-trivial action on the moduli space. We show that under an embedding of these gauge theories into string theory, the Hecke duality groups are represented by T-duality transformations.Comment: 8 pages, latex. v2: references adde

Contact symmetry of time-dependent Schr\"odinger equation for a two-particle system: symmetry classification of two-body central potentials

Symmetry classification of two-body central potentials in a two-particle Schr\"{o}dinger equation in terms of contact transformations of the equation has been investigated. Explicit calculation has shown that they are of the same four different classes as for the point transformations. Thus in this problem contact transformations are not essentially different from point transformations. We have also obtained the detailed algebraic structures of the corresponding Lie algebras and the functional bases of invariants for the transformation groups in all the four classes

Conformal windows of SP(2N) and SO(N) gauge theories from topological excitations on R3 * S1

We derive an estimate of the lower boundary of the conformal window of SP(2N) and SO(N) gauge theories with fermionic matter in several different representations. We calculate the index of topological excitations for these groups on the manifold R3 * S1, from which we deduce the scale of the generation of the mass gap of the theory. This is then used to approximate the critical value of the number of species for the onset of conformality on R4. We also provide a detailed comparison with other estimates of the conformal window.Comment: 23 pages, 4 figures, 7 tables; updated results, added comparisons, references adde

On fundamental domains and volumes of hyperbolic Coxeter-Weyl groups

We present a simple method for determining the shape of fundamental domains of generalized modular groups related to Weyl groups of hyperbolic Kac-Moody algebras. These domains are given as subsets of certain generalized upper half planes, on which the Weyl groups act via generalized modular transformations. Our construction only requires the Cartan matrix of the underlying finite-dimensional Lie algebra and the associated Coxeter labels as input information. We present a simple formula for determining the volume of these fundamental domains. This allows us to re-produce in a simple manner the known values for these volumes previously obtained by other methods.Comment: v2: to be published in Lett Math Phys (reference added, typo corrected

Generators of simple Lie algebras in arbitrary characteristics

In this paper we study the minimal number of generators for simple Lie algebras in characteristic 0 or p > 3. We show that any such algebra can be generated by 2 elements. We also examine the 'one and a half generation' property, i.e. when every non-zero element can be completed to a generating pair. We show that classical simple algebras have this property, and that the only simple Cartan type algebras of type W which have this property are the Zassenhaus algebras.Comment: 26 pages, final version, to appear in Math. Z. Main improvements and corrections in Section 4.

Lattice Point Generating Functions and Symmetric Cones

We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically contrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out the specific cases of a symmetry group of type A (previously known) and types B and D (new). We obtain several applications of the special cases in type B, including identities involving permutation statistics and lecture hall partitions.Comment: 19 page

Modular Lie algebras and the Gelfand-Kirillov conjecture

Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. We show that if the Gelfand-Kirillov conjecture holds for g, then g has type A_n, C_n or G_2.Comment: 20 page

Diagonalization of an Integrable Discretization of the Repulsive Delta Bose Gas on the Circle

We introduce an integrable lattice discretization of the quantum system of n bosonic particles on a ring interacting pairwise via repulsive delta potentials. The corresponding (finite-dimensional) spectral problem of the integrable lattice model is solved by means of the Bethe Ansatz method. The resulting eigenfunctions turn out to be given by specializations of the Hall-Littlewood polynomials. In the continuum limit the solution of the repulsive delta Bose gas due to Lieb and Liniger is recovered, including the orthogonality of the Bethe wave functions first proved by Dorlas (extending previous work of C.N. Yang and C.P. Yang).Comment: 25 pages, LaTe

Periodic harmonic functions on lattices and points count in positive characteristic

This survey addresses pluri-periodic harmonic functions on lattices with values in a positive characteristic field. We mention, as a motivation, the game "Lights Out" following the work of Sutner, Goldwasser-Klostermeyer-Ware, Barua-Ramakrishnan-Sarkar, Hunzikel-Machiavello-Park e.a.; see also 2 previous author's preprints for a more detailed account. Our approach explores harmonic analysis and algebraic geometry over a positive characteristic field. The Fourier transform allows us to interpret pluri-periods of harmonic functions on lattices as torsion multi-orders of points on the corresponding affine algebraic variety.Comment: These are notes on 13p. based on a talk presented during the meeting "Analysis on Graphs and Fractals", the Cardiff University, 29 May-2 June 2007 (a sattelite meeting of the programme "Analysis on Graphs and its Applications" at the Isaac Newton Institute from 8 January to 29 June 2007