Quadratic algebras and dynamics in curved spaces. II. The Kepler problem

The symmetry aspects of the Kepler problem in a space of constant negative curvature are considered. It is shown that the algebra of the hidden symmetry reduces to the quadratic Racah algebraQR(3), and this makes it possible to express the coefficients of the overlapping of the wave functions in the spherical and parabolic coordinates in terms of Wilson-Racah polynomials. It is shown that the dynamical symmetry algebra that generates the spectrum is the quadratic Jacobi algebraQJ(3). Its ladder operators permit explicit construction of wave functions in the coordinate representation with the ground state as the starting point.Donetsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 396–410, June, 1992.     

Quadratic spaces and quaternion algebras

It is shown that the existence of a nontrivial quaternion algebra over a field F of characteristic not equal to 2 implies the existence of quadratic F[X₁...,X _n ]-spaces (n2) which are not extensions of F.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 110–120, 1978.     

Quadratic algebras with Ext algebras generated in two degrees

Green and Marcos (2005) [2] call a graded k-algebra δ-Koszul if the corresponding Yoneda algebra is finitely generated and there exists a function δ:N→N such that is zero if j≠δ(i). For any integer m≥3 we exhibit a noncommutative quadratic δ-Koszul algebra for which the Yoneda algebra is generated in degrees (1,1) and (m,m+1). These examples answer a question of Green and Marcos. These algebras are not Koszul but m-Koszul (in the sense of Backelin).     

Fraction-dense algebras and spaces

A fraction-dense (semi-prime) commutative ring A with 1 is one for which the classical quotient ring is rigid in its maximal quotient ring. The fraction-dense f-rings are characterized as those for which the space of minimal prime ideals is compact and extremally disconnected. For Archimedean lattice-ordered groups with this property it is shown that the Dedekind and order completion coincide. Fraction-dense spaces are defined as those for which C (X) is fraction-dense. If X is compact, then this notion is equivalent to the coincidence of the absolute of X and its quasi-F cover.     

Automorphically rigid group algebras. I. Semisimple algebras

Classes of automorphically rigid semisimple group algebras KG are defined over finite fields K and finite groups G.     

Boundary dimension in negatively curved spaces

LetX be a negatively curved (Gromov hyperbolic) space. We construct a bound on dim X when a group of isometries acts cocompactly onX. We construct an example of a negatively curved space with infinite-dimensional boundary.     

Weyl product algebras and modulation spaces

We discuss algebraic properties of the Weyl product acting on modulation spaces. For a certain class of weight functions ω we prove that is an algebra under the Weyl product if p∈[1,∞] and 1?q?min(p,p^′). For the remaining cases p∈[1,∞] and min(p,p^′)<q?∞ we show that the unweighted spaces Mp,q are not algebras under the Weyl product.     

Transitive algebras and reductive algebras on reproducing analytic Hilbert spaces

In this paper, we consider the well-known transitive algebra problem and reductive algebra problem on vector valued reproducing analytic Hilbert spaces. For an analytic Hilbert space H(k) with complete Nevanlinna-Pick kernel k, it is shown that both transitive algebra problem and reductive algebra problem on multiplier invariant subspaces of H(k)⊗Cm have positive answer if the algebras contain all analytic multiplication operators. This extends several known results on the problems.     

Complex function algebras and removable spaces

The compact Hausdorff space X has the Complex Stone-Weierstrass Property (CSWP) iff it satisfies the complex version of the Stone-Weierstrass Theorem. W. Rudin showed that all scattered spaces have the CSWP. We describe some techniques for proving that certain non-scattered spaces have the CSWP. In particular, if X is the product of a compact ordered space and a compact scattered space, then X has the CSWP if and only if X does not contain a copy of the Cantor set.     

Quadratic variation of martingales in Riesz spaces

We derive quadratic variation inequalities for discrete-time martingales, sub- and supermartingales in the measure-free setting of Riesz spaces. Our main result is a Riesz space analogue of Austin?s sample function theorem, on convergence of the quadratic variation processes of martingales.     

Semi — Quadratic sets in projective spaces

The purpose of this paper is to characterize semi-quadrics in projective spacesP of finite dimension 2 at least. A concept of semi-quadratic set inP is introduced: a semi-quadratic setQ inP is essentially a set of points ofP such that the union of all tangent lines at each pointp ofQ is either a hyperplane ofP orP itself. (A tangent line ofQ atp is a line contained inQ or meetingQ exactly inp). The main result is that a semi-quadratic set which is invariant under “many” perspectivities is a semi-quadric.     

Almost-classical Lie algebras. I

Let g be the Lie algebra of a semi-simple algebraic group G over an algebraically closed field k of characteristic p>0. In the first part of this paper we study the structure of g, and focus on the deviations from the characteristic 0 case. In the second part we determine, for almost-simple G, the automorphism group of g.     

Quadratic spaces over finite fields and codes

Let V be a finite-dimensional quadratic space over a finite field GF(?) of characteristic different from 2. It is shown that, even if V is singular, the geometry of V is completely determined by the number of points on the unit sphere, the “sphere of the nonsquares,” and the “0-sphere.” For ? = 3, this implies that two codes over GF(3) with the same weight enumerator are isometric.