scholarly journals INVERTIBLE BIMODULES, MIYASHITA ACTION IN MONOIDAL CATEGORIES AND AZUMAYA MONOIDS

2016 ◽  
Vol 225 ◽  
pp. 1-63
Author(s):  
ALESSANDRO ARDIZZONI ◽  
LAIACHI EL KAOUTIT
Keyword(s):  
Algebraic Group ◽  
Algebraic Variety ◽  
Classical Theory ◽  
Classical Result ◽  
Symmetric Case ◽  
Monoidal Categories ◽  
Special Cases ◽  
Common Framework ◽  
Hopf Algebroids ◽  
Geometric Nature

In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automorphisms. In the symmetric case, we show that for certain Azumaya monoids, which are abundant in practice, the corresponding Miyashita action is always an isomorphism of groups. This generalizes Miyashita’s classical result and sheds light on other applications of geometric nature which cannot be treated using the classical theory. In order to illustrate our methods, we give a concrete application to the category of comodules over commutative (flat) Hopf algebroids. This obviously includes the special cases of split Hopf algebroids (action groupoids), which for instance cover the situation of the action of an affine algebraic group on an affine algebraic variety.

scholarly journals Complete reducibility: variations on a theme of Serre

2021 ◽  
Author(s):  
Maike Gruchot ◽  
Alastair Litterick ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Spherical Building ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
General Properties ◽  
Complete Reducibility ◽  
Special Cases ◽  
Variations On A Theme

AbstractIn this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and $$\sigma $$ σ -complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

ALGEBRO-GEOMETRIC INVARIANTS OF GROUPS (THE DIMENSION SEQUENCE OF A REPRESENTATION VARIETY)

2011 ◽  
Vol 21 (04) ◽  
pp. 595-614 ◽  
Author(s):  
S. LIRIANO ◽  
S. MAJEWICZ
Keyword(s):  
Algebraic Group ◽  
Free Group ◽  
Algebraic Variety ◽  
Commutator Subgroup ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Geometric Invariants ◽  
Torus Knot ◽  
Irreducible Components ◽  
Irreducible Variety

If G is a finitely generated group and A is an algebraic group, then RA(G) = Hom (G, A) is an algebraic variety. Define the "dimension sequence" of G over A as Pd(RA(G)) = (Nd(RA(G)), …, N0(RA(G))), where Ni(RA(G)) is the number of irreducible components of RA(G) of dimension i (0 ≤ i ≤ d) and d = Dim (RA(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofFn = 〈x1, …, xn〉, and letG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety andV-1 = {ρ | ρ ∈ RSL(2, ℂ)(Fn), ρ(w) = -I} ≠ ∅, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). Theorem B.Let w be a nontrivial word in the free group on{x1, …, xn}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). We also show that if G = 〈x1, . ., xn, y; W = yp〉, where p ≥ 1 and W is a word in Fn = 〈x1, …, xn〉, and A = PSL(2, ℂ), then Dim (RA(G)) = Max {3n, Dim (RA(G′)) +2 } ≤ 3n + 1 for G′ = 〈x1, …, xn; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; xp = yt〉 then Pd(RSL(2, ℂ)(G))≠Pd(RPSL(2, ℂ)(G)).

scholarly journals Are Ratings the Worst Form of Credit Assessment Except for All the Others?

2018 ◽  
Vol 53 (1) ◽  
pp. 299-334 ◽  
Author(s):  
Andreas Blöchlinger ◽  
Markus Leippold
Keyword(s):  
Theoretical Model ◽  
Prediction Model ◽  
Incomplete Information ◽  
Transformation Method ◽  
Combined Approach ◽  
Powerful Predictor ◽  
Publicly Traded ◽  
Special Cases ◽  
Common Framework ◽  
Distance To Default

We present a prediction model to forecast corporate defaults. In a theoretical model, under incomplete information in a market with publicly traded equity, we show that our approach must outperform ratings, Altman’sZ-score, and Merton’s distance to default. We reconcile the statistical and structural approaches under a common framework; that is, our approach nests Altman’s and Merton’s approaches as special cases. Empirically, the combined approach is indeed the most powerful predictor, and the numbers of observed defaults align well with the estimated probabilities. With a new transformation method, we obtain cycle-adjusted forecasts that still outperform ratings.

scholarly journals The scheme of Lie sub-algebras of a Lie algebra and the equivariant cotangent map

1974 ◽  
Vol 53 ◽  
pp. 59-70
Author(s):  
William J. Haboush
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Variety ◽  
Local Properties

The main object of this paper is to develop techniques for investigating the local properties of actions of an algebraic group on an algebraic variety. Our main tools are certain schemes which may be associated to Lie algebras.

scholarly journals The fundamental equations of quantum mechanics

1925 ◽  
Vol 109 (752) ◽  
pp. 642-653 ◽  
Keyword(s):  
Classical Theory ◽  
Correspondence Principle ◽  
Classical Mechanics ◽  
Stationary States ◽  
Special Cases ◽  
Restricted Region ◽  
The Right ◽  
Fundamental Equations ◽  
Limiting Case ◽  
Mathematical Operations

It is well known that the experimental facts of atomic physics necessitate a departure from the classical theory of electrodynamics in the description of atomic phenomena. This departure takes the form, in Bohr’s theory, of the special assumptions of the existence of stationary states of an atom, in which it does not radiate, and of certain rules, called quantum conditions, which fix the stationary states and the frequencies of the radiation emitted during transitions between them. These assumptions are quite foreign to the classical theory, but have been very successful in the interpretation of a restricted region of atomic phenomena. The only way in which the classical theory is used is through the assumption that the classical laws hold for the description of the motion in the stationary states, although they fail completely during transitions, and the assumption, called the Correspondence Principle, that the classical theory gives the right results in the limiting case when the action per cycle of the system is large compared to Planck’s constant h , and in certain other special cases. In a recent paper Heisenberg puts forward a new theory, which suggests that it is not the equations of classical mechanics that are in any way at fault, but that the mathematical operations by which physical results are deduced from them require modification. All the information supplied by the classical theory can thus be made use of in the new theory.

On a Family of Distributions obtained from Orbits

1986 ◽  
Vol 38 (1) ◽  
pp. 179-214 ◽  
Author(s):  
James Arthur
Keyword(s):  
Conjugacy Class ◽  
Algebraic Group ◽  
Difficult Case ◽  
Reductive Algebraic Group ◽  
Section 8 ◽  
Special Cases ◽  
Image Position ◽  
The Right ◽  
Family Of Distributions ◽  
Opposite Extreme

Suppose that G is a reductive algebraic group defined over a number field F. The trace formula is an identityof distributions. The terms on the right are parametrized by “cuspidal automorphic data”, and are defined in terms of Eisenstein series. They have been evaluated rather explicitly in [3]. The terms on the left are parametrized by semisimple conjugacy classes and are defined in terms of related G(A) orbits. The object of this paper is to evaluate these terms.In previous papers we have already evaluated in two special cases. The easiest case occurs when corresponds to a regular semisimple conjugacy class in G(F). We showed in Section 8 of [1] that for such an , could be expressed as a weighted orbital integral over the conjugacy class of σ. (We actually assumed that was “unramified”, which is slightly more general.) The most difficult case is the opposite extreme, in which corresponds to {1}.

scholarly journals NONEXTREME BLACK HOLES FROM INTERSECTING M-BRANES

1998 ◽  
Vol 13 (08) ◽  
pp. 1305-1328 ◽  
Author(s):  
NOBUYOSHI OHTA ◽  
TAKASHI SHIMIZU
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Higher Dimensions ◽  
Symmetric Case ◽  
Spherically Symmetric ◽  
Field Equations ◽  
Deformation Parameters ◽  
Special Cases ◽  
Two Parameter ◽  
Black Hole Solutions

We investigate the possibility of extending nonextreme black hole solutions made of intersecting M-branes to those with two nonextreme deformation parameters, similar to Reissner–Nordstrøm solutions. General analysis of possible solutions is carried out to reduce the problem of solving field equations to a simple algebraic one for static spherically-symmetric case in D dimensions. The results are used to show that the extension to two-parameter solutions is possible for D= 4,5 dimensions but not for higher dimensions, and that the area of horizon always vanishes in the extreme limit for black hole solutions for D≥6 except for two very special cases which are identified. Various solutions are also summarized.

scholarly journals Semisimplification for subgroups of reductive algebraic groups

2020 ◽  
Vol 8 ◽  
Author(s):  
Michael Bate ◽  
Benjamin Martin ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Invariant Theory ◽  
Algebraic Groups ◽  
Geometric Invariant ◽  
Reductive Algebraic Group ◽  
Complete Reducibility ◽  
Special Cases ◽  
Closed Fields ◽  
Completely Reducible ◽  
Algebraically Closed Fields

Let G be a reductive algebraic group—possibly non-connected—over a field k, and let H be a subgroup of G. If $G= {GL }_n$ , then there is a degeneration process for obtaining from H a completely reducible subgroup $H'$ of G; one takes a limit of H along a cocharacter of G in an appropriate sense. We generalise this idea to arbitrary reductive G using the notion of G-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a G-completely reducible subgroup $H'$ of G, unique up to $G(k)$ -conjugacy, which we call a k-semisimplification of H. This gives a single unifying construction that extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL _n$ and with Serre’s ‘G-analogue’ of semisimplification for subgroups of $G(k)$ from [19]). We also show that under some extra hypotheses, one can pick $H'$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf, and Rousseau.

scholarly journals Reconciling Novelty and Complexity through a Rational Analysis of Curiosity

2019 ◽  
Author(s):  
Rachit Dubey ◽  
Tom Griffiths
Keyword(s):  
Causal Structure ◽  
Experimental Results ◽  
Rational Agent ◽  
Rational Behavior ◽  
Rational Analysis ◽  
Computational Problem ◽  
Special Cases ◽  
Wide Range ◽  
Common Framework ◽  
Intermediate Degree

Curiosity is considered to be the essence of science and an integral component of cognition. What prompts curiosity in a learner? Previous theoretical accounts of curiosity remain divided – novelty-based theories propose that new and highly uncertain stimuli pique curiosity whereas complexity-based theories propose that stimuli with an intermediate degree of uncertainty stimulate curiosity. In this article, we present a rational analysis of curiosity by considering the computational problem underlying curiosity, which allows us to model these distinct accounts of curiosity in a common framework. Our approach posits that a rational agent should explore stimuli that maximally increase the usefulness of its knowledge and that curiosity is the mechanism by which humans approximate this rational behavior. Critically, our analysis shows that the causal structure of the environment can determine whether curiosity is driven by either highly uncertain or moderately uncertain stimuli. This suggests that previous theories need not be in contention but are special cases of a more general account of curiosity. Experimental results confirm our predictions and demonstrate that our theory explains a wide range of findings about human curiosity, including its subjectivity and malleability.

scholarly journals A note on the mean value theorem for special homogeneous spaces

1996 ◽  
Vol 143 ◽  
pp. 111-117 ◽  
Author(s):  
Masanori Morishita ◽  
Takao Watanabe
Keyword(s):  
Algebraic Group ◽  
Algebraic Variety ◽  
Homogeneous Spaces ◽  
Rational Numbers ◽  
Rational Points ◽  
Linear Algebraic Group ◽  
Mean Value ◽  
Mean Value Theorem ◽  
The Mean

Let G be a connected linear algebraic group and X an algebraic variety, both defined over Q, the field of rational numbers. Suppose that G acts on X transitively and the action is defined over Q. Suppose that the set of rational points X(Q) is non-empty. Choosing x ∈ X(Q) allows us to identify G/Gx and X as varieties over Q, there Gx is the stabilizer of x.

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