A Note on the Categorification of Lie Algebras

Goyvaerts, Isar; Vercruysse, Joost

doi:10.1007/978-4-431-54270-4_41

Cited by 38 publications

(16 citation statements)

References 12 publications

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“…Nouns and verbs with a frequency of mention greater than 120 were selected to ensure that they were produced by a significant number of speakers. Second, these words were input to the Powergrep program (Goyvaerts, 2013) to determine how many participants produced each of these words. Only lexical items that were used by more than 50% of the 120 healthy participants were selected as stimuli for rating.…”

Section: Methodsmentioning

confidence: 99%

Effects of context and word class on lexical retrieval in Chinese speakers with anomic aphasia

et al. 2014

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Background Differences in processing nouns and verbs have been investigated intensely in psycholinguistics and neuropsychology in past decades. However, the majority of studies examining retrieval of these word classes have involved tasks of single word stimuli or responses. While the results have provided rich information for addressing issues about grammatical class distinctions, it is unclear whether they have adequate ecological validity for understanding lexical retrieval in connected speech which characterizes daily verbal communication. Previous investigations comparing retrieval of nouns and verbs in single word production and connected speech have reported either discrepant performance between the two contexts with presence of word class dissociation in picture naming but absence in connected speech, or null effects of word class. In addition, word finding difficulties have been found to be less severe in connected speech than picture naming. However, these studies have failed to match target stimuli of the two word classes and between tasks on psycholinguistic variables known to affect performance in response latency and/or accuracy. Aims The present study compared lexical retrieval of nouns and verbs in picture naming and connected speech from picture description, procedural description, and story-telling among 19 Chinese speakers with anomic aphasia and their age, gender, and education matched healthy controls, to understand the influence of grammatical class on word production across speech contexts when target items were balanced for confounding variables between word classes and tasks. Methods & Procedures Elicitation of responses followed the protocol of the AphasiaBank consortium (http://talkbank.org/AphasiaBank/). Target words for confrontation naming were based on well-established naming tests, while those for narrative were drawn from a large database of normal speakers. Selected nouns and verbs in the two contexts were matched for age-of-acquisition (AoA) and familiarity. Influence of imageability was removed through statistical control. Outcomes & Results When AoA and familiarity were balanced, nouns were retrieved better than verbs, and performance was higher in picture naming than connected speech. When imageability was further controlled for, only the effect of task remained significant. Conclusions The absence of word class effects when confounding variables are controlled for is similar to many previous reports; however, the pattern of better word retrieval in naming is rare but compatible with the account that processing demands are higher in narrative than naming. The overall findings have strongly suggested the importance of including connected speech tasks in any language assessment and evaluation of language rehabilitation of individuals with aphasia.

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Section: Methodsmentioning

confidence: 99%

Effects of context and word class on lexical retrieval in Chinese speakers with anomic aphasia

et al. 2014

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“…The upshot of this is the notion of a quasi-Frobenius Lie object, which can be viewed as the analogue of a Frobenius object in the current setting. The starting point for this particular step is the categorification of Lie algebra due to Goyvaerts and Vercruysse [12]:…”

Section: G-quasi-frobenius Lie Algebrasmentioning

confidence: 99%

“…To obtain this formulation, we introduce the notion of a quasi-Frobenius Lie object for any additive symmetric monoidal category. The work of Goyvaerts and Vercuysse on the categorification of Lie algebras [12] provides the foundation for defining quasi-Frobenius Lie objects. The latter then yields an alternate (yet equivalent) definition of a g-quasi-Frobenius Lie algebra: a g-quasi Frobenius Lie algebra is simply a quasi Frobenius Lie object in Rep(g), where Rep(g) is the category of finite dimensional representations of g. Using the categorical formulation of [18] as motivation, we obtain the Lie version of a G-Frobenius algebra: for a fixed finite dimensional Lie bialgebra (g, γ), the Lie version of a G-Frobenius algebra is a quasi-Frobenius Lie object in Rep(D(g)).…”

Section: Introductionmentioning

confidence: 99%

$\mathfrak{g}$-quasi-Frobenius Lie algebras

Pham¹

2016

Arch. Math. (Brno)

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A Lie version of Turaev's G-Frobenius algebras from 2dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a g-quasi-Frobenius Lie algebra for g a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra (q, β) together with a left g-module structure which acts on q via derivations and for which β is g-invariant.Geometrically, g-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic Lie groups with an action by a Lie group G which acts via symplectic Lie group automorphisms. In addition to geometry, g-quasi-Frobenius Lie algebras can also be motivated from the point of view of category theory. Specifically, g-quasi Frobenius Lie algebras correspond to quasi Frobenius Lie objects in Rep(g). If g is now equipped with a Lie bialgebra structure, then the categorical formulation of G-Frobenius algebras given in [18] suggests that the Lie version of a G-Frobenius algebra is a quasi-Frobenius Lie object in Rep(D(g)), where D(g) is the associated (semiclassical) Drinfeld double. We show that if g is a quasitriangular Lie bialgebra, then every g-quasi-Frobenius Lie algebra has an induced D(g)-action which gives it the structure of a D(g)-quasi-Frobenius Lie algebra.

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“…Suppose now that C and D are additive, symmetric monoidal categories, and that R : C → D is an additive symmetric monoidal functor with left adjoint L. As known (see e.g. [6]) an additive symmetric monoidal functor preserves Lie algebras. Hence, we obtain a new functor…”

Section: 2mentioning

confidence: 99%

On the duality of generalized Lie and Hopf algebras

Goyvaerts

Vercruysse

2014

Advances in Mathematics

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Abstract. We show how, under certain conditions, an adjoint pair of braided monoidal functors can be lifted to an adjoint pair between categories of Hopf algebras. This leads us to an abstract version of Michaelis' theorem, stating that given a Hopf algebra H, there is a natural isomorphism of Lie algebras Q(H) * ∼ = P (H • ), where Q(H) * is the dual Lie algebra of the Lie coalgebra of indecomposables of H, and P (H • ) is the Lie algebra of primitive elements of the Sweedler dual of H. We apply our theory to Turaev's Hopf group-(co)algebras.

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A Note on the Categorification of Lie Algebras

Abstract: Abstract. In this short note we study Lie algebras in the framework of symmetric monoidal categories. After a brief review of the existing work in this field and a presentation of earlier studied and new examples, we examine which functors preserve the structure of a Lie algebra.

Cited by 38 publications

References 12 publications

Effects of context and word class on lexical retrieval in Chinese speakers with anomic aphasia

Effects of context and word class on lexical retrieval in Chinese speakers with anomic aphasia

$\mathfrak{g}$-quasi-Frobenius Lie algebras

On the duality of generalized Lie and Hopf algebras

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