ON THE REDUCED NORM 1 GROUP OF A DIVISION ALGEBRA OVER A GLOBAL FIELD

AbstractLet D be a finite-dimensional division algebra over its center and R = D[t;σ,δ] a skew polynomial ring. Under certain assumptions on δ and σ, the ring of central quotients D(t;σ,δ) = {f/g|f ∈ D[t;σ,δ],g ∈ C(D[t;σ,δ])} of D[t;σ,δ] is a central simple algebra with reduced norm N. We calculate the norm N(f) for some skew polynomials f ∈ R and investigate when and how the reducibility of N(f) reflects the reducibility of f.

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On Projective Modules and Automorphisms of Central Separable Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-005-6 ◽

1969 ◽

Vol 21 ◽

pp. 44-53 ◽

Cited By ~ 5

Author(s):

L. N. Childs

Keyword(s):

Division Algebra ◽

Quaternion Algebra ◽

Central Simple Algebra ◽

Simple Algebra ◽

Algebraic Number Field ◽

Projective Modules ◽

Rank One ◽

Central Simple ◽

Separable Algebras ◽

Reduced Norm

This paper developed from, and complements, the paper by F. R. DeMeyer (see 6).In the first section of this paper we note a correspondence between projective modules of a central separable R-algebra A and the two-sided ideals of central separable algebras in the same class as A in the Brauer group of R. When R has the property that rank one projective A-modules are free, this correspondence yields a bijection between isomorphism types of indecomposable projective A-modules and the isomorphism types of algebras in the Brauer class of A which are the analogue of division algebra components in the field case. This bijection was remarked on without proof by DeMeyer in (6).Pursuing the ideas behind this correspondence, we consider the situation for a separable order A in a central simple algebra A over an algebraic number field, and obtain, by means of results involving the reduced norm, a generalization of DeMeyer's remark except when the division algebra component of A is a totally definite quaternion algebra (Theorem 3.3).

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Indices for the Exceptional Bruhat-Tits Buildings

10.23943/princeton/9780691166902.003.0036 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Capital Letter ◽

Coxeter Diagram ◽

Root Group ◽

Relative Type ◽

Quaternion Division Algebra

This chapter considers the affine Tits indices for exceptional Bruhat-Tits buildings. It begins with a few small observations and some notations dealing with the relative type of the affine Tits indices, the canonical correspondence between the circles in a Tits index and the vertices of its relative Coxeter diagram, and Moufang sets. It then presents a proposition about an involutory set, a quaternion division algebra, a root group sequence, and standard involution. It also describes Θ‎-orbits in S which are disjoint from A and which correspond to the vertices of the Coxeter diagram of Ξ‎ and hence to the types of the panels of Ξ‎. Finally, it shows how it is possible in many cases to determine properties of the Moufang set and the Tits index for all exceptional Bruhat-Tits buildings of type other than Latin Capital Letter G with Tilde₂.

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Existence

10.23943/princeton/9780691166902.003.0016 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Structure Theorem ◽

Quadratic Extension ◽

Quadratic Space ◽

Division Algebras ◽

Separable Extension ◽

Valued Fields ◽

Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

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Totally Wild Quadratic Forms of Type E7

10.23943/princeton/9780691166902.003.0015 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Quadratic Space ◽

Trace Map ◽

Quaternion Division Algebra

This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, and the trace and trace map. The chapter also considers two other propositions: the first states that if the trace map is not equal to zero, then the Moufang residues R₀ and R₁ are not indifferent; the second states that if the trace map is equal to zero, then the Moufang residues R₀ and R₁ are both indifferent.

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Linked Tori, II

10.23943/princeton/9780691166902.003.0006 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Integral Domain ◽

Quadratic Space ◽

Small Observation ◽

Quaternion Division Algebra

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

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Synaptic Molecular and Neurophysiological Markers Are Independent Predictors of Progression in Alzheimer’s Disease

Journal of Alzheimer s Disease ◽

10.3233/jad-201234 ◽

2021 ◽

pp. 1-12

Author(s):

Una Smailovic ◽

Ingemar Kåreholt ◽

Thomas Koenig ◽

Nicholas J. Ashton ◽

Bengt Winblad ◽

...

Keyword(s):

Alzheimer’S Disease ◽

Alzheimer's Disease ◽

Global Field ◽

Prognostic Indicators ◽

Early Event ◽

Functional Markers ◽

Beta Band ◽

Global Field Power ◽

Synaptic Markers ◽

Eeg Recordings

Background: Cerebrospinal fluid (CSF) neurogranin and quantitative electroencephalography (qEEG) are potential molecular and functional markers of synaptic pathology in Alzheimer’s disease (AD). Synaptic markers have emerged as candidate prognostic indicators of AD since synaptic degeneration was shown to be an early event and the best correlate of cognitive deficits in patients along the disease continuum. Objective: The present study investigated the association between CSF neurogranin and qEEG measures as well as their potential to predict clinical deterioration in mild cognitive impairment (MCI) patients. Methods: Patients diagnosed with MCI (n = 99) underwent CSF conventional AD biomarkers and neurogranin analysis and resting-state EEG recordings. The study population was further stratified into stable (n = 41) and progressive MCI (n = 31), based on the progression to AD dementia during two years follow-up. qEEG analysis included computation of global field power and global field synchronization in four conventional frequency bands. Results: CSF neurogranin levels were associated with theta power and synchronization in the progressive MCI group. CSF neurogranin and qEEG measures were significant predictors of progression to AD dementia, independent of baseline amyloid status in MCI patients. A combination of CSF neurogranin with global EEG power in theta and global EEG synchronization in beta band exhibited the highest classification accuracy as compared to either of these markers alone. Conclusion: qEEG and CSF neurogranin are independent predictors of progression to AD dementia in MCI patients. Molecular and neurophysiological synaptic markers may have additive value in a multimodal diagnostic and prognostic approach to dementia.

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Between interpretation and the subject: Revisiting Bakhtin’s theory of polyphony

Semiotica ◽

10.1515/sem-2019-0086 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Jie Zhang ◽

Hongbing Yu

Keyword(s):

Literary Theory ◽

Global Field ◽

Availability Bias ◽

Obvious Consequence ◽

The Subject ◽

Influential Theory

AbstractThis paper affords a critical and historical reappraisal of Bakhtin’s theory of polyphony. It addresses the issue of the subjectivity of interpretation in the reception and formulation of this highly influential theory in literary semiotics. Following a revaluation of three major patterns of interpretation of polyphony that have emerged in the global field of literary theory since 1929, as well as Bakhtin’s shift in emphasis in 1963, we find that Bakhtin’s theorizing of polyphony, based on his seemingly inconsistent interpretation of Dostoevsky’s novels, was defined by his own subjectivity as well. An obvious consequence of such subjective predispositions in both the reception of Bakhtin’s theory and his own treatment of Dostoevsky’s polyphonic novels is that they have instigated a type of perpetuating availability bias in approaching the theory of polyphony. This revelation is key to understanding the wholeness of the theory of polyphony from a diachronic perspective. By tracing the cultural and intellectual sources of Dostoevsky’s polyphonic creation, this paper attempts to reframe and restore the Bakhtinian idea of polyphony to its fullness, which we believe can be encapsulated in one phrase: harmony without uniformity.

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