On composition and absolute-valued algebras

2006 ◽  
Vol 136 (4) ◽  
pp. 717-731 ◽  
Author(s):  
José Antonio Cuenca Mira
Keyword(s):  
Finite Dimension ◽  
Quadratic Algebra ◽  
Central Idempotent ◽  
Quadratic Algebras ◽  
Large Classes ◽  
Anisotropic Norm ◽  
Composition Algebras ◽  
Characteristic 2 ◽  
The Absolute

In this note we give a complete description of the composition algebras A over fields of characteristic ≠ 2, 3 in the following cases: if A has an anisotropic norm and x2x = xxx2 for every element; when A has a unitary central idempotent, it satisfies the identity (x2x2)x2 = x2(x2x2), and A is of finite dimension or has anisotropic norm. As a consequence, we obtain the existence, up to an isomorphism, of only seven absolute-valued algebras with a non-zero central idempotent where the last identity holds. This result completes the study of the absolute-valued algebras of this kind that was initiated by El-Mallah and Agawany.We also introduce the class of e-quadratic algebra, which contains the quadratic algebras, but also includes large classes of composition and absolute-valued algebras. Many results on composition, absolute-valued and e-quadratic algebras are shown, and new proofs of some well-known theorems are given.

Solution of the Dirac equation with some superintegrable potentials by the quadratic algebra approach

2014 ◽  
Vol 29 (06) ◽  
pp. 1450028 ◽  
Author(s):  
S. Aghaei ◽  
A. Chenaghlou
Keyword(s):  
Dirac Equation ◽  
Quadratic Algebra ◽  
Two Dimensional ◽  
Oscillator Algebra ◽  
Quadratic Algebras ◽  
Energy Eigenvalues ◽  
Vector Potentials ◽  
Algebra Approach ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

The Dirac equation with scalar and vector potentials of equal magnitude is considered. For the two-dimensional harmonic oscillator superintegrable potential, the superintegrable potentials of E8 (case (3b)), S4 and S2, the Schrödinger-like equations are studied. The quadratic algebras of these quasi-Hamiltonians are derived. By using the realization of the quadratic algebras in a deformed oscillator algebra, the structure function and the energy eigenvalues are obtained.

scholarly journals Quadratic algebra contractions and second-order superintegrable systems

2014 ◽  
Vol 12 (05) ◽  
pp. 583-612 ◽  
Author(s):  
Ernest G. Kalnins ◽  
W. Miller
Keyword(s):  
Constant Curvature ◽  
Two Dimensions ◽  
Second Order ◽  
Quadratic Algebra ◽  
Quadratic Algebras ◽  
Physical Systems ◽  
Special Cases ◽  
Superintegrable Systems ◽  
Wilson Polynomials ◽  
Symmetry Algebras

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of second-order superintegrable systems in two dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. For constant curvature spaces, we show that the free quadratic algebras generated by the first- and second-order elements in the enveloping algebras of their Euclidean and orthogonal symmetry algebras correspond one-to-one with the possible superintegrable systems with potential defined on these spaces. We describe a contraction theory for quadratic algebras and show that for constant curvature superintegrable systems, ordinary Lie algebra contractions induce contractions of the quadratic algebras of the superintegrable systems that correspond to geometrical pointwise limits of the physical systems. One consequence is that by contracting function space realizations of representations of the generic superintegrable quantum system on the 2-sphere (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.

ON A LOCAL ANALOGUE OF THE GROTHENDIECK CONJECTURE

2000 ◽  
Vol 11 (02) ◽  
pp. 133-175
Author(s):  
VICTOR A. ABRASHKIN
Keyword(s):  
Galois Group ◽  
Absolute Galois Group ◽  
Residue Fields ◽  
Characteristic 2 ◽  
The Absolute

We prove that the functor from the category of all complete discrete valuation fields with finite residue fields of characteristic ≠2 to the category of profinite filtered groups given by taking the Galois group of corresponding field together with its filtration by higher ramification subgroups is fully faithful. If [K; ℚp]<∞ we also study the opportunity to recover K from the knowledge of the filtered group ΓK(p)/ΓK(p)(a), where a>0, ΓK(p) is the absolute Galois group of the maximal p-extension of K and filtration is induced by ramification filtration.

scholarly journals SUBALGEBRAS OF GOLOD–SHAFAREVICH ALGEBRAS

2009 ◽  
Vol 19 (03) ◽  
pp. 423-442 ◽  
Author(s):  
THOMAS VODEN
Keyword(s):  
Associative Algebra ◽  
Quadratic Algebra ◽  
Quadratic Algebras

A graded associative algebra generated by m elements of degree one is called Golod–Shafarevich (GS) if it is presented with less than m2/4 relators of degree at least two. We explore conditions under which subalgebras of graded GS algebras are themselves GS. We prove that infinitely many Veronese powers of an algebra presented by m generators and r relators are GS if [Formula: see text]. For quadratic algebras, the bound is improved to [Formula: see text]. We also show that if A is a generic quadratic algebra presented by m generators and r relators, then all Veronese powers of A are GS if [Formula: see text], and all but finitely many Veronese powers of A are not GS if [Formula: see text].

Simple Classical Lie Algebras in Characteristic 2 and Their Gradations, II

2014 ◽  
Vol 21 (02) ◽  
pp. 207-214
Author(s):  
Alexandre N. Grishkov ◽  
Marinês Guerreiro
Keyword(s):  
Lie Algebras ◽  
Finite Dimension ◽  
Type B ◽  
Simple Lie Algebras ◽  
Characteristic 2

This paper is a continuation of [2]. We prove Conjecture 5.1 of [2] which gives a characterization of simple Lie algebras of finite dimension of type B2ℓ, C2ℓ, D2ℓ+1, E7 and E8 in terms of some gradations of these algebras over a field of characteristic 2.

scholarly journals A Classification of Some Quadratic Algebras

2006 ◽  
Vol 13 (01) ◽  
pp. 133-148
Author(s):  
Heather C. McGilvray
Keyword(s):  
Quadratic Algebra ◽  
Quadratic Algebras ◽  
Linear Resolution

We investigate a group of quadratic algebras and the associated resolutions of the field. When the quadratic algebra is Koszul, we provide the associated linear resolution; when not Koszul, we describe the maps of the resolution up to the point of non-linearity. In all cases, the resolutions are shown to be surprisingly regular and quickly stabilizing.

scholarly journals Conjugate polynomials over quadratic algebras

1992 ◽  
Vol 52 (2) ◽  
pp. 154-174
Author(s):  
Timothy Stokes
Keyword(s):  
Commutative Algebra ◽  
Euclidean Geometry ◽  
Commutative Rings ◽  
Closed Field ◽  
Quadratic Algebra ◽  
Two Dimensional ◽  
Close Correspondence ◽  
Quadratic Algebras ◽  
Natural Operation ◽  
Geometry Theorem Proving

AbstractThis paper gives variants of results from classical algebraic geometry and commutative algebra for quadratic algebras with conjugation. Quadratic algebras are essentially two-dimensional algebras with identity over commutative rings with identity, on which a natural operation of conjugation may be defined. We define the ring of conjugate polynomials over a quadratic algebra, and define c-varieties. In certain cases a close correspondence between standard varieties and c-varieties is demonstrated, and we establish a correspondence between conjugate and standard polynomials, which leads to variants of the Hilbert Nullstellensatz if the commutativering is an algebraically closed field. These results may be applied to automated Euclidean geometry theorem proving.

scholarly journals KOSZUL CALCULUS

2017 ◽  
Vol 60 (2) ◽  
pp. 361-399 ◽  
Author(s):  
ROLAND BERGER ◽  
THIERRY LAMBRE ◽  
ANDREA SOLOTAR
Keyword(s):  
Duality Theorem ◽  
Derived Categories ◽  
Quadratic Algebra ◽  
Koszul Duality ◽  
True Nature ◽  
Quadratic Algebras ◽  
Koszul Homology ◽  
Cup Products ◽  
Koszul Cohomology

AbstractWe present a calculus that is well-adapted to homogeneous quadratic algebras. We define this calculus on Koszul cohomology – resp. homology – by cup products – resp. cap products. The Koszul homology and cohomology are interpreted in terms of derived categories. If the algebra is not Koszul, then Koszul (co)homology provides different information than Hochschild (co)homology. As an application of our calculus, the Koszul duality for Koszul cohomology algebras is proved foranyquadratic algebra, and this duality is extended in some sense to Koszul homology. So, the true nature of the Koszul duality theorem is independent of any assumption on the quadratic algebra. We compute explicitly this calculus on a non-Koszul example.

Bulk standards for low-temperature biological x-ray microanalysis

1984 ◽  
Vol 42 ◽  
pp. 282-283
Author(s):  
P. Echlin ◽  
M. McKoon ◽  
E.S. Taylor ◽  
C.E. Thomas ◽  
K.L. Maloney ◽  
...  
Keyword(s):  
Phase Separation ◽  
Low Temperature ◽  
Analytical Method ◽  
Salt Solutions ◽  
Absolute Concentration ◽  
Cooling Process ◽  
X Ray ◽  
The Absolute ◽  
Analytical Procedures ◽  
Electrical Conductors

Although sections of frozen salt solutions have been used as standards for x-ray microanalysis, such solutions are less useful when analysed in the bulk form. They are poor thermal and electrical conductors and severe phase separation occurs during the cooling process. Following a suggestion by Whitecross et al we have made up a series of salt solutions containing a small amount of graphite to improve the sample conductivity. In addition, we have incorporated a polymer to ensure the formation of microcrystalline ice and a consequent homogenity of salt dispersion within the frozen matrix. The mixtures have been used to standardize the analytical procedures applied to frozen hydrated bulk specimens based on the peak/background analytical method and to measure the absolute concentration of elements in developing roots.

A Quantitative Ultrastructural Study of Peripheral Blood Lymphocytes Containing Parallel Tubular Arrays in Epstein-Barr Virus and Cytomegalo-Virus Mononucleosis

1981 ◽  
Vol 39 ◽  
pp. 454-455
Author(s):  
C. M. Payne ◽  
P. M. Tennican
Keyword(s):  
Peripheral Blood ◽  
Lymphocyte Count ◽  
Epstein Barr Virus ◽  
Absolute Lymphocyte Count ◽  
Peripheral Blood Lymphocytes ◽  
Clinical State ◽  
Barr Virus ◽  
The Absolute ◽  
Epstein Barr ◽  
Tubular Arrays

In the normal peripheral circulation there exists a sub-population of lymphocytes which is ultrastructurally distinct. This lymphocyte is identified under the electron microscope by the presence of cytoplasmic microtubular-like inclusions called parallel tubular arrays (PTA) (Figure 1), and contains Fc-receptors for cytophilic antibody. In this study, lymphocytes containing PTA (PTA-lymphocytes) were quantitated from serial peripheral blood specimens obtained from two patients with Epstein -Barr Virus mononucleosis and two patients with cytomegalovirus mononucleosis. This data was then correlated with the clinical state of the patient.It was determined that both the percentage and absolute number of PTA- lymphocytes was highest during the acute phase of the illness. In follow-up specimens, three of the four patients' absolute lymphocyte count fell to within normal limits before the absolute PTA-lymphocyte count.In one patient who was followed for almost a year, the absolute PTA- lymphocyte count was consistently elevated (Figure 2). The estimation of absolute PTA-lymphocyte counts was determined to be valid after a morphometric analysis of the cellular areas occupied by PTA during the acute and convalescent phases of the disease revealed no statistical differences.

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