Classification Of Extensions Research Articles

Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel

The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well known result. A remarkable generalisation of this equivalence, due to Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admit an action of the structure group by automorphisms. This paper proves the existence of suitably equivariant transition functions for such groupoids, generalising consequently the classification of principal bundles by means of their transition functions, to extensions of principal bundles by an equivariant form of \v{C}ech cohomology.

When Are Broader Brands Stronger Brands? An Accessibility Perspective on the Success of Brand Extensions

It is common for brands to extend into additional product categories. The most successful extensions involve brands that are associated with benefits that are valued in the extension category. We propose that brand extension success also depends on the accessibility of these benefit associations and that accessibility, in turn, depends on the amount of interference by competing brand associations (e.g., category associations). One implication of this proposition is that broad brands (i.e., brands offering a portfolio of diverse products) will tend to have more accessible benefit associations than narrow brands (i.e., brands offering a portfolio of similar products) and can therefore engage in more successful brand extensions than narrow brands, even when the narrow brands are more similar to the extension category. However, when benefit associations are equally accessible and diagnostic, the evaluation of brand extensions will instead be dictated by the similarity between brand and extension category associations.

Contribution to the classification of minimal extensions

We show that certain quasi-Banach spaces arising from quasi-linear functions on ℓ 1 are not isomorphic. We also show that if Γ is a large set, then every quasi-linear function on ℓ 1( Γ) is near to a true linear map on some infinite dimensional subspace.

Fusion category algebras

The fusion system F on a defect group P of a block b of a finite group G over a suitable p -adic ring O does not in general determine the number l ( b ) of isomorphism classes of simple modules of the block. We show that conjecturally the missing information should be encoded in a single second cohomology class α of the constant functor with value k × on the orbit category F c of F -centric subgroups Q of P of b which “glues together” the second cohomology classes α ( Q ) of Aut F (Q) with values in k × in Külshammer–Puig [Invent. Math. 102 (1990) 17–71]. We show that if α exists, there is a canonical quasi-hereditary k -algebra F (b) such that Alperin's weight conjecture becomes equivalent to the equality l(b)=l( F (b)) . By work of C. Broto, R. Levi, B. Oliver [J. Amer. Math. Soc. 16 (2003) 779–856], the existence of a classifying space of the block b is equivalent to the existence of a certain extension category L of F c by the center functor Z . If both invariants α , L exist, we show that there is an O -algebra L (b) associated with b having F (b) as quotient such that Alperin's weight conjecture becomes again equivalent to the equality l(b)=l( L (b)) ; furthermore, if b has an abelian defect group, L (b) is isomorphic to a source algebra of the Brauer correspondent of b .

Fusion category algebras

The fusion system F on a defect group P of a block b of a finite group G over a suitable p-adic ring O does not in general determine the number l( b) of isomorphism classes of simple modules of the block. We show that conjecturally the missing information should be encoded in a single second cohomology class α of the constant functor with value k × on the orbit category F c of F -centric subgroups Q of P of b which “glues together” the second cohomology classes α( Q) of Aut F (Q) with values in k × in Külshammer–Puig [Invent. Math. 102 (1990) 17–71]. We show that if α exists, there is a canonical quasi-hereditary k-algebra F (b) such that Alperin's weight conjecture becomes equivalent to the equality l(b)=l( F (b)) . By work of C. Broto, R. Levi, B. Oliver [J. Amer. Math. Soc. 16 (2003) 779–856], the existence of a classifying space of the block b is equivalent to the existence of a certain extension category L of F c by the center functor Z . If both invariants α, L exist, we show that there is an O -algebra L(b) associated with b having F (b) as quotient such that Alperin's weight conjecture becomes again equivalent to the equality l(b)=l( L(b)) ; furthermore, if b has an abelian defect group, L(b) is isomorphic to a source algebra of the Brauer correspondent of b.

Module categories of simple current extensions of vertex operator algebras

We study module categories of simple current extensions of rational C 2-cofinite vertex operator algebras of CFT-type and prove that they are semisimple. We also develop a method of induced modules for the simple current extensions.

ℤ[] is Euclidean

AbstractWe provide the first unconditional proof that the ring ℤ[] is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of ℚ. It is proved that if K is a real quadratic field (modulo the existence of two special primes of K) or if K is a cyclotomic extension of ℚ then:the ring of integers of K is a Euclidean domain if and only if it is a principal ideal domain.The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when K is a totally real extension of degree at least three. The main changes are a new Motzkintype lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property.

Obstruction Theory for Extensions of Categorical Groups

For any categorical group H, we introduce the categorical group Out(H) and then the well-known group exact sequence 1 → Z(H ) → H → Aut(H ) → Out(H ) → 1i s raised to a categorical group level by using a suitable notion of exactness. Breen's Schreier theory for extensions of categorical groups is codified in terms of homomorphism to Out(H) and then we develop a sort of Eilenberg-Mac Lane obstruction theory that solves the general problem of the classification of all categorical group extensions of a group G by a categorical group H, in terms of ordinary group cohomology. Mathematics Subject Classifications (2000): 18D10, 18G50, 20J06.

Factors Influencing Successful Brand Extensions

Organisations frequently follow brand extension strategies. This paper investigates the impact of category similarity, brand reputation, perceived risk and consumer innovativeness on the success of brand extensions in FMCG, durable goods and services sectors. A set of hypotheses were developed and tested in a study amongst 701 consumers. The findings show that extensions into categories more similar to the original brand tend to be more readily accepted. Likewise, the reputation of the original brand is an important factor influencing the success of the extension. These findings are consistent across FMCG, durable goods and services brands. However, perceived risk about the extension category was only found to enhance acceptability of extensions for durable goods and services brands. Innovative consumers are more positively disposed towards service brand extensions than FMCG and durable goods brand extensions.

Logical Construction of Final Coalgebras

We prove that every finitary polynomial endofunctor of a category C has a final coalgebra, provided that C is locally Cartesian closed, it has finite coproducts and is an extensive category, it has a natural number object.

Family Pediatrics

Family Pediatrics

Continuous fields of [formula omitted]-algebras arising from extensions of tensor [formula omitted]-categories

The notion of extension of a given C ∗ -category C by a C ∗ -algebra A is introduced. In the commutative case A=C(Ω), the objects of the extension category are interpreted as fiber bundles over Ω of objects belonging to the initial category. It is shown that the Doplicher–Roberts algebra (DR-algebra in the following) associated to an object in the extension of a strict tensor C ∗ -category is a continuous field of DR-algebras coming from the initial one. In the case of the category of the hermitian vector bundles over Ω the general result implies that the DR-algebra of a vector bundle is a continuous field of Cuntz algebras. Some applications to Pimsner C ∗ -algebras are given.

Central Extensions of Stephenson's Algebras

This paper completes the classification of central extensions of three dimensional Artin-Schelter regular algebras to four dimensional Artin-Schelter regular algebras. Let A be an AS regular algebra of global dimension three and let D be an extension of A by a central graded element z, i.e., D/⟨z⟩ = A. If A is generated by elements of degree one, those algebras D which are again AS regular have been classified in Le Bruyn et al. (Le Bruyn L., Smith, S. P., Van den Bergh, M. (1996). Central extensions of three dimensional Artin-Schelter regular algebras. Math. Zeitschrift 222:171–212.) and Cassidy (Cassidy, T. (1999). Global dimension 4 extensions of Artin-Schelter regular algebras. J. Algebra 220:225–254.). If A is not generated by elements of degree one, then A falls under a classification due to Stephenson (Stephenson, D. R. (1996). Artin-Schelter regular algebras of global dimension three. J. Algebra 183(1):55–73 and Stephenson, D. R. (1997). Algebras associated to elliptic curves. Trans. Amer. Math. Soc. 349(6):2317–2340.). We classify the AS regular central extensions of Stephenson's algebras by proving that the regularity of D and z is equivalent to the regularity of z in low degree and this is equivalent to easily verifiable conditions on the defining relations for D.

Classification of Novikov Complete Logics with Extra Logical Constants

We give an exhaustive classification of Novikov complete extensions of the intuitionistic propositional logic in a language with extra logical constants.

“Deep” and “Surface” Cues: Brand Extension Evaluations by Children and Adults

We present a conceptual as well as empirical analysis showing that 11–12-year-old children, relative to adults, evaluate brand extensions by relying more on surface cues (e.g., brand name characteristics used to launch the extension) and less on deep cues (e.g., category similarity between the parent brand and the extension category). In experiment 1, children gave equivalent evaluations of brand extensions regardless of category similarity (e.g., Coca-Cola extending to iced tea vs. toffee), whereas adults rated near extensions (e.g., iced tea) more favorably than far extensions (e.g., toffee). In experiment 2, children evaluated near extensions more favorably than far extensions when they were cued to make similarity judgments prior to evaluation but rated near and far extensions equivalently when they were not cued prior to evaluation. In experiment 3, children based their evaluations on the extension name's linguistic characteristics regardless of category similarity, whereas adults based their evaluations on category similarity regardless of name characteristics. Children rated extensions with a rhyming name (e.g., “Coca-Cola Gola” iced tea; “Wrigley's Higley” toffee) more positively than extensions with a nonrhyming name (e.g., “Coca-Cola Higley” iced tea; “Wrigley's Gola” toffee), whereas adults rated them similarly. Theoretical implications on branding and categorization research as well as managerial implications are discussed.

Derived and Stable Equivalence Classification of Twisted Multifold Extensions of Piecewise Hereditary Algebras of Tree Type

Derived and Stable Equivalence Classification of Twisted Multifold Extensions of Piecewise Hereditary Algebras of Tree Type

Extensions with Almost Maximal Depth of Ramification

The paper is devoted to classification of finite abelian extensions L/K which satisfy the condition [L:K]|D L/K. Here K is a complete discretely valued field of characteristic 0 with an arbitrary residue field of prime characteristic p, D L/K is the different of L/K. This condition means that the depth of ramification in L/K has its “almost maximal” value. The condition appeared to play an important role in the study of additive Galois modules associated with the extension L/K. The study is based on the use of the notion of independently ramified extensions, introduced by the authors. Two principal theorems are proven, describing almost maximally ramified extensions in the cases when the absolute ramification index e is (resp. is not) divisible by p-1. Bibliography: 7 titles.

Brand Extensions: Prestige Brand Effects

Competition and consumer brand attitudes make introducing a new brand risky and very expensive, increasing the attractiveness of brand extensions. Brand extension research has established that original brand quality perceptions, product category fit, consumer perceptions of the product category complementarity and substitutability, plus the transferability of design and manufacturing capability significantly influence extension attitude formation. However, these studies fail to differentiate between brand types and have aggregated out category effects. Using a national survey we show that brand type (prestige, functional) and extension category are important variables in consumer brand extension attitude formation. Perceived quality influences functional brands more than prestige brands as prestige brand attitudes assume quality. Functional brands experience less dilution than prestige brands, implying less risk and greater expandability than argued by previous authors.

Do We Really Know how Consumers Evaluate Brand Extensions? Empirical Generalizations Based on Secondary Analysis of Eight Studies

The authors investigate the empirical generalizability of Aaker and Keller's model of how consumers evaluate brand extensions. Various replications have reported different results. Using a comprehensive data set containing the data from the original study and seven replications conducted around the world, the authors undertake a secondary analysis to understand what generalizations emerge. The study has implications for the understanding of how brand extensions are evaluated and how empirical generalizations are made. For brand extensions, Aaker and Keller's model hypothesizes that evaluations of brand extensions are based on the quality of the original brand, the fit between the parent and extension categories, and the interaction of the two. The authors find support for this full model despite published results, including Aaker and Keller's own, that support only some of the hypotheses. The authors find evidence that the level of contribution of each of these components varies by brand and culture. With respect to empirical generalizations, the key implication is that it is premature to make firm conclusions about theory on the basis of only one study.

When do completion processes give rise to extensive categories?

We consider various (free) completion processes: the exact completion and the regular completion of a category with weak finite limits, the pre-regular completion of a category with finite products and weak finite limits, the exact completion of a regular category, the regular reflection of a pre-regular category, and the filtered-colimit completion of a small category. In each case we give necessary and sufficient conditions for the completion to be extensive; or, in the case of the pre-regular completion, for the completion to satisfy a weakened notion of extensivity which we call pre-extensivity.