Geometrical approach to the free sovable groups

We give a topological interpretation of the free metabelian group, following the plan described in [Ver1,Ver2]. This interpretation is based on considering the Caley graph of a finitely generated group G as one-dimensional complex; its homology group with integer coefficients as G-space; and corresponding extension of the homology group determinated by the canonical cocycle. This gives a recursive approach to free solvable groups of any degree of solvability. For metabelian case we calculate second cohomology and describe all satellite groups.Comment: 14 page

Complex and CR-structures on compact Lie groups associated to Abelian actions

It was shown by Samelson and Wang that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui who have also given a complete algebraic description of these structures. In this article we present an alternative and more geometric construction of this type of invariant structures on a compact Lie group K when it is semisimple. We prove that each left-invariant complex structure, or each CR-structure of maximal dimension with a transverse CR-action by R, is induced by a holomorphic C^l action on a quasi-projective manifold X naturally associated to K. We then show that X admits more general Abelian actions, also inducing complex or CR structures on K which are generically non-invariant

Quantum Geometry and the Planck Scale

We consider some general aspects of the new noncommutative or quantum geometry coming out of the theory of quantum groups, in connection with Planck scale physics. A generalisation of Fourier or wave-particle duality on curved spaces emerges. Another feature is the need for particles with fractional or braid statistics. The conformal group also has a special role.Comment: 12 pages Latex, no figure

Fundamental groups of complements of curves as solvable groups

We discuss the applications of fundamental groups (of complements of curves) computations (and possibly the computations of the second homotopy group as a model over it) to the classification of algebraic surface. We prove that the fundamental group of the complement of the branch curve of a generic projection of a Veronese surface to the complex plane is an "almost solvable" group in the sense that it contains a solvable group of finite index and thus we can consider the second fundamental group as model over the first.Comment: To be published in IMCP 9 (Proceedings of Hirzebruch 65-birthday

Algebraic Geometry and Physics

This article is an interdisciplinary review and an on-going progress report over the last few years made by myself and collaborators in certain fundamental subjects on two major theoretic branches in mathematics and theoretical physics: algebraic geometry and quantum physics. I shall take a practical approach, concentrating more on explicit examples rather than formal developments. Topics covered are divided in three sections: (I) Algebraic geometry on two-dimensional exactly solvable statistical lattice models and its related Hamiltonians: I will report results on the algebraic geometry of rapidity curves appeared in the chiral Potts model, and the algebraic Bethe Ansatz equation in connection with quantum inverse scattering method for the related one-dimensional Hamiltonion chain, e.g., XXZ, Hofstadter type Hamiltonian. (II) Infinite symmetry algebras arising from quantum spin chain and conformal field theory: I will explain certain progress made on Onsager algebra, the relation with the superintegrable chiral Potts quantum chain and problems on its spectrum. In conformal field theory, mathematical aspects of characters of N=2 superconformal algebra are discussed, especially on the modular invariant property connected to the theory. (III). Algebraic geometry problems on orbifolds stemming from string theory: I will report recent progress on crepant resolutions of quotient singularity of dimension greater than or equal to three. The direction of present-day research of engaging finite group representations in the geometry of orbifolds is briefly reviewed, and the mathematical aspect of various formulas on the topology of string vacuum will be discussed.Comment: 18 pages, Latex; Talk presented in The Third Asian Mathematical Conference, October 23-27, 2000 at Manila, Philippines; references updated, minor typos correcte

Isospin from Spin by Compositenes

We propose a new method to generate the internal isospin degree of freedom by non-local bound states. This can be seen as motivated by Bargmann-Wigner like considerations, which originated from local spin coupling. However, our approach is not of purely group theoretical origin, but emerges from a geometrical model. The rotational part of the Lorentz group can be seen to mutate into the internal iso-group under some additional assumptions. The bound states can thereafter be characterized by either a triple of spinors (\xi_1, \xi_2, \eta) or a pair of an average spinor and a ``gauge'' transformation (\phi, R). Therefore, this triple can be considered to be an isospinor. Inducing the whole dynamics from the covariant gauge coupling we arrive at an isospin gauge theory and its Lagrangian formulation. Clifford algebraic methods, especially the Hestenes approach to the geometric meaning of spinors, are the most useful concepts for such a development. The method is not restricted to isospin, which served as an example only.Comment: 8 pages, 21kb, 1 figure, uses sprocl.st

The Casimir Invariants of Universal Lie algebra extensions via commutative structures

We consider the Casimir Invariants related to some a special kind of Lie-algebra extensions, called universal extensions. We show that these invariants can be studied using the equivalence between the universal extensions and the commutative algebras and consider in detail the so called coextension structures, arising in the calculation of the Casimir functions.Comment: LaTex, no figure

Lattice models and N=2N=2 supersymmetry

We review the construction of exactly solvable lattice models whose continuum limits are $N=2$ supersymmetric models. Both critical and off-critical models are discussed. The approach we take is to first find lattice models with natural topological sectors, and then identify the continuum limits of these sectors with topologically twisted $N=2$ supersymmetric field theories. From this, we then describe how to recover the complete lattice versions of the $N=2$ supersymmetric field theories. We discuss a number of simple physical examples and we describe how to construct a broad class of models. We also give a brief review of the scattering matrices for the excitations of these models. (Contribution to the procedings of the Cargese meeting on ``String Theory, Conformal Models and Topological Field Theories'', May 12-21, 1993.)Comment: 49 pages, 12 figures, Latex, USC preprint 93/02

The Word Problem for the Singular Braid Monoid

We give a solution to the word problem for the singular braid monoid SB_n. The complexity of the algorithm is quadratic in the product of the word length and the number of the singular generators in the word. Furthermore we algebraically reprove a result of Fenn, Keyman and Rourke that the monoid embeds into a group and we compute the cohomological dimension of this group.Comment: 18 pages, 4 figure

Lie-algebras and linear operators with invariant subspaces

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of some algebra of differential (difference) operators in finite-dimensional representation plus an operator annihilating the finite-dimensional invariant subspace. In low dimensions a classification is given by algebras $sl_2({\bold R})$ (for differential operators in ${\bold R}$) and $sl_2({\bold R})_q$ (for finite-difference operators in ${\bold R}$), $osp(2,2)$ (operators in one real and one Grassmann variable, or equivalently, $2 \times 2$ matrix operators in ${\bold R}$), $sl_3({\bold R})$, $sl_2({\bold R}) \oplus sl_2({\bold R})$ and $gl_2 ({\bold R}) \ltimes {\bold R}^{r+1}\ , r$ a natural number (operators in ${\bold R^2}$). A classification of linear operators possessing infinitely many finite-dimensional invariant subspaces with a basis in polynomials is presented. A connection to the recently-discovered quasi-exactly-solvable spectral problems is discussed.Comment: 47pp, AMS-LaTe