On the Embedding of Analytic Moufang Loops in Alternative Algebras

Mathematical Notes -     

Moufang loops and alternative algebras

Let be the algebra of classical real octonions or the (split) algebra of octonions over the finite field 2$">. Then the quotient loop of the Moufang loop of all invertible elements of the algebra modulo its center is not embedded into a loop of invertible elements of any alternative algebra.

     

Sugawara Construction for the Birepresentation of the Moufang Loop

We show explicitly the connection between the infinitesimal form of the birepresentation of the Moufang loop and the octonionic representation of SO(8). Possible types of Sugawara construction for the birepresentations are considered.     

Zorn's Matrices and Finite Index Subloops#

Zorn's Algebra ?(R) has a multiplicative function called determinant with properties similar to the usual one. The set of elements in ?(R) with determinant 1 is a Moufang loop that we will denote by IΓ. In our main result we prove that if R is a Dedekind algebraic number domain that contains a unit of infinite order, each finite index subloop ?, such that IΓ has the weak Lagrange property relative to ?, is a congruence subloop.     

例谈数形结合在高数解题中的应用

通过微分中值的等式证明、求渐近线方程、定积分中的换元法、几何图形的描绘以及曲线积分的计算等例题,说明将代数运算或证明与几何直观相结合给解高等数学问题带来的好处．     

The Structure of Frobenius Algebras and Separable Algebras

We present a unified approach to the study of separable and Frobenius algebras. The crucial observation is that both types of algebras are related to the nonlinear equation R¹²R²³=R²³R¹³=R¹³R¹², called the FS-equation. Solutions of the FS-equation automatically satisfy the braid equation, an equation that is in a sense equivalent to the quantum Yang–Baxter equation. Given a solution to the FS-equation satisfying a certain normalizing condition, we can construct a Frobenius algebra or a separable algebra A(R) – the normalizing condition is different in both cases. The main result of this paper is the structure of these two fundamental types of algebras: a finite dimensional Frobenius or separable k-algebra A is isomorphic to such an A(R). A(R) can be described using generators and relations. A new characterization of Frobenius extensions is given: B A is Frobenius if and only if A has a B-coring structure (A, , ) such that the comultiplication : A A_B A is an A-bimodule map.     

The Ideal Property and Traces

We prove a characterization of the ideal property for a large class of C*–algebras in terms of vanishing conditions for the finite traces on the algebra. An application of this result to a class of inductive limit C*–algebras (GAH algebras) is also given.     

Finitistic dimension and Igusa–Todorov algebras

The notion of Igusa–Todorov algebras is introduced in connection with the (little) finitistic dimension conjecture, and the conjecture is proved for those algebras. Such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds, e.g., algebras with the representation dimension at most 3, algebras with radical cube zero, monomial algebras and left serial algebras, etc. It is an open question whether all artin algebras are Igusa–Todorov. We provide some methods to construct many new classes of (2-)Igusa–Todorov algebras and thus obtain many algebras such that the finitistic dimension conjecture holds. In particular, we show that the class of 2-Igusa–Todorov algebras is closed under taking endomorphism algebras of projective modules. Hence, if all quasi-hereditary algebras are 2-Igusa–Todorov, then all artin algebras are 2-Igusa–Todorov by [V. Dlab, C.M. Ringel, Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring, Proc. Amer. Math. Soc. 107 (1) (1989) 1–5] and have finite finitistic dimension.     

Cartan–Slodkowski spectra, splitting elements and noncommutative spectral mapping theorems

In this paper, we prove Slodkowski version of the infinite-dimensional spectral mapping theorem and Cartan–Slodkowski version of the finite-dimensional spectral mapping theorem for nilpotent operator Lie subalgebras with respect to the various noncommutative functional calculi.     

Auslander-Regular and Cohen–Macaulay Quantum Groups

The quantized enveloping C(q)-algebra U _q (C) associated to a Cartan matirx C is Auslander-regular and Cohen–Macaulay. This is deduced from a general theorem, which also applies to solvable polynomial algebras. The results are obtained by constructing a new filtration keeping the properties of the associated graded algebra of a given multi-filtered algebra.     

Rigorous Formulation of a 2D Conformal Model in the Fock–Krein Space: Construction of the Global Op J *-Algebra of Fields and Currents

We use the formalism of the 2D massless scalar field model in an indefinite space of the Fock–Krein type as a basis for constructing a rigorous formulation of 2D quantum conformal theories. We show that the sought construction is a several-stage procedure whose central block is the construction of a new type of representation of the Virasoro algebra. We develop the first stage of this procedure, which is to construct a special global algebra of fields and currents generated by exponential generators. We obtain a system of commutation relations for the Wick-squared currents used in the definition of the Virasoro generators. We prove the existence of Wick exponentials of the current given by operator-valued generalized functions; the sought global algebra is rigorously defined as the algebra of current and field, Wick and normal exponentials on a common dense invariant domain in a Fock–Krein space.     

Coxeter Transformations in Quantum Groups

It is natural to define a Coxeter transformation in quantum groups by the product of Lusztig's symmetries in some order. In this note, we show that the Ringel–Hall algebra approach enables us to determine the behavior of its action in case of finite type.     

BRST-Invariant Algebra of Constraints in Terms of Commutators and Quantum Antibrackets

We establish the general structure of the BRST-invariant algebra of constraints in its commutator and antibracket forms via the formulation of algebra-generating equations in a supplementally extended phase space. New ghost-type variables behave as fields and antifields with respect to quantum antibrackets. The explicit form of the BRST-invariant gauge algebra is given in detail for rank-one theories with a Weyl- and a Wick-ordered ghost sector. We construct a fixed-gauge unitarizing Hamiltonian and show that the formalism is physically equivalent to the standard BRST–BFV approach.     

The univalent Bloch–Landau constant, harmonic symmetry and conformal glueing

By modifying a domain first suggested by Ruth Goodman in 1945 and by exploiting the explicit solution by Fedorov of the Pólya–Chebotarev problem in the case of four symmetrically placed points, an improved upper bound for the univalent Bloch–Landau constant is obtained. The domain that leads to this improved bound takes the form of a disk from which some arcs are removed in such a way that the resulting simply connected domain is harmonically symmetric in each arc with respect to the origin. The existence of domains of this type is established using techniques from conformal welding, and some general properties of harmonically symmetric arcs in this setting are established.     

Virtual Knots and Infinite-Dimensional Lie Algebras

In the present work, we construct an invariant of virtual knots valued in (infinite-dimensional) Lie Algebras and establish some properties of it. This leads to some heuristic ideas how to construct quandles and extract (virtual) link invariants.     

Trees, Renormalization and Differential Equations

The Butcher group and its underlying Hopf algebra of rooted trees were originally formulated to describe Runge–Kutta methods in numerical analysis. In the past few years, these concepts turned out to have far-reaching applications in several areas of mathematics and physics: they were rediscovered in noncommutative geometry, they describe the combinatorics of renormalization in quantum field theory. The concept of Hopf algebra is introduced using a familiar example and the Hopf algebra of rooted trees is defined. Its role in Runge–Kutta methods, renormalization theory and noncommutative geometry is described.     

A note on Tutte polynomials and Orlik–Solomon algebras

Let be a (central) arrangement of hyperplanes in and the dependence matroid of the linear forms . The Orlik–Solomon algebra of a matroid is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The graded algebra is isomorphic to the cohomology algebra of the manifold . The Tutte polynomial is a powerful invariant of the matroid . When is a rank 3 matroid and the θ_Hi are complexifications of real linear forms, we will prove that determines . This result partially solves a conjecture of Falk.     

A Partnership Approach to Improving Student Attitudes About Sharks and Scientists

This article describes the methods and impact of a student–teacher–scientist research partnership on student attitudes. The partnership objective was to teach students about the diverse roles of sharks in the marine environment while personally connecting students with scientific study. Students (N = 229) participated in lessons about shark biology and helped the partnering scientist design experimental protocols and analyze data. A self‐selected subset of students also volunteered (n = 82) for a field component working with live hammerhead sharks (Sphyrna lewinii). Student surveys before and after the partnership suggested that negative attitudes about sharks are due largely to lack of exposure, and direct attention to students' stereotypes about sharks resulted in significant attitude improvement. Change in students' attitudes toward scientists, however, was minimal. Students' negative views of scientists did decline significantly, but their overall views of scientists were relatively positive to begin with. Also of interest was the students' unremitting association of scientists with specialized equipment and the students' lack of personal connection to scientific ways of examining the world, suggesting that partnerships may be more effective at personally connecting students with scientific process if they explicitly incorporate activities designed to improve students' view of themselves as scientists.     

Torsion in the cohomology of finite loop spaces and the Eilenberg–Moore spectral sequence

We use the Eilenberg–Moore spectral sequence to study torsion phenomena in the integral cohomology of finite loop spaces with maximal torus generalizing some results for compact Lie groups due to Kac.     

Weyl transforms associated with the Hankel transform in Clifford analysis

In this paper, we define the Hankel–Wigner transform in Clifford analysis and therefore define the corresponding Weyl transform. We present some properties of this kind of Hankel–Wigner transform, and then give the criteria of the boundedness of the Weyl transform and compactness on the L^p space. Copyright © 2005 John Wiley & Sons, Ltd.